Bubble Testing under Deterministic Trends∗

Xiaohu WangChinese University of Hong Kong

Jun YuSingapore Management University

September 9, 2019

Abstract

This paper develops the asymptotic theory of the ordinary least squares estimatorof the autoregressive (AR) coeffi cient in various AR models, when data is generatedfrom trend-stationary models in different forms. It is shown that, depending on howthe autoregression is specified, the commonly used right-tailed unit root tests maytend to reject the null hypothesis of unit root in favor of the explosive alternative. Anew procedure to implement the right-tailed unit root tests is proposed. It is shownthat when the data generating process is trend-stationary, the test statistics basedon the proposed procedure cannot find evidence of explosiveness. Whereas, whenthe data generating process is mildly explosive, the unit root tests find evidence ofexplosiveness. Hence, the proposed procedure enables robust bubble testing underdeterministic trends. Empirical implementation of the proposed procedure using datafrom the stock and the real estate markets in the US reveals some interesting findings.While our proposed procedure flags the same number of bubbles episodes in the stockdata as the method developed in Phillips, Shi and Yu (2015a, PSY), the estimatedtermination dates by the proposed procedure match better with the data. For realestate data, all negative bubble episodes flagged by PSY are no longer regarded asbubbles by the proposed procedure.

JEL classification: C12, C22, G01Keywords: Autoregressive regressions, right-tailed unit root test, explosive and mildlyexplosive processes, deterministic trends, coeffi cient-based statistic, t-statistic.

1 Introduction

Other than its catastrophic effect on global financial markets of many kinds, the recentglobal financial crisis also turned the field of economics, including economic theory offinancial markets, econometric theory of financial time series, and economic policies, onits head. Acknowledging an important empirical observation that often a financial bubble

∗Xiaohu Wang, Department of Economics, The Chinese University of Hong Kong, Shatin, N.T.,Hong Kong. Email: xiaohu.wang@cuhk.edu.hk. Jun Yu, School of Economics and Lee Kong ChianSchool of Business, Singapore Management University, 90 Stamford Road, Singapore 178903. Email: yu-jun@smu.edu.sg. Wang acknowledges the support from the Hong Kong Research Grants Council GeneralResearch Fund under No. 14503718.

1

precedes immediately a financial crisis, a great deal of efforts that have been made ineconometrics is to test for the presence of bubbles in financial time series and to estimatethe origination date and the termination date of each bubble in real time. Widely usedmethods which have achieved great success in detecting and dating periodically collapsingbubbles are the recursive right-tailed unit root testing procedures proposed in Phillips,Wu and Yu (2011), Phillips and Yu (2011), and Phillips, Shi and Yu (2015a, 2016b,PSY hereafter). Harvey, et al. (2016), and Harvey, Leybourne and Zu (2017a, 2017b)extended the procedures to deal with the case with heteroscedasticity. A recent surveyand comparisons of alternative methods in bubble testing and dating can be found inHomm and Breitung (2012).

In an environment of time-invariant discount rate, the standard no-arbitrage conditionimplies that

pt = ft + bt, (1)

where pt is an asset price at time t, ft :=∞∑i=1

(1+r)−iEt(dt+i) is a “fundamental”component

with dt+i representing the payment received over the period from t+ i− 1 to t+ i due tothe ownership of the asset, r is the discount rate (r > 0), and bt is a bubble componentwhich satisfies

Et(bt+1) = (1 + r)bt. (2)

Since β := 1 + r > 1, when bt > 0, Et(bt+1) = βbt > bt, indicating the presence of anupward explosive behavior in bt+1, i.e. a positive bubble. When ft+1 does not involve anyexplosive behavior, the upward explosive behavior in bt+1 suggests that pt+1 manifests inan explosive positive bubble behavior. When bt < 0, Et(bt+1) = βbt < bt, indicating thepresence of a downward explosive behavior in bt+1, i.e. a negative bubble. When ft+1

does not involve any explosive behavior, the downward explosive behavior in bt+1 leads toan explosive negative bubble behavior in pt+1. In this case pt+1 could still be positive aslong as ft+1 > −bt+1.

The testing and dating method for bubbles relies on the technique of fitting to timeseries data (i.e. prices adjusted by fundamental values) the following autoregressive (AR)model where the AR coeffi cient (β) may take a different value in a bubble (or even a crisis)period from that in a normal period:

yt = α+ βyt−1 + εt. (3)

To test H0 : β = 1 against H1 : β > 1, both the t-test and the coeffi cient-based test havebeen used. Both tests are based on the ordinary least squares (LS) estimate of β.1 Whenthe tests are implemented recursively, they can detect when the time series switches froma unit root model to an explosive model, and vice versa.

PSY (2015a) applied the proposed recursive method to a long time series data on themonthly S&P 500 stock price index-dividend ratio over the period from January 1871

1More recent studies have proposed methods based on the weighted LS estimate to deal with het-eroscedasticity; see, for example, Harvey et al. (2017a).

2

Figure 1: The price-dividend ratio of S&P500, the backward SADF of PSY, and the 95%critical value between January 1871 to December 2010.

to December 2010. Figure 1 reproduces PSY’s Figure 7 where the original data, thesequence of the sup t-statistics, and the sequence of the 95% finite sample critical valuesare plotted. According to PSY, whenever the test statistic exceeds the critical value, itindicates a bubble episode. As a consequence, several well-known bubble episodes in thisseries have been identified, including the dot-com bubble (1995M11—2001M08).

Interestingly, the PSY strategy also identifies two periods of market downturns as bub-ble episodes, namely the 1917 stock market crash (1917M08-1918M04) and the subprimemortgage crisis (2009M02-M04). PSY conjectured that “the identification of crashes asbubbles may be caused by very rapid changes in the data”. Some recent studies, suchas Phillips and Shi (2017) and Harvey, et al. (2016, 2017b), modelled downturns usinga stationary AR process with β in Model (3) smaller than 1. However, a stationary ARmodel is at odd with the testing result found in PSY which suggests β > 1. Instead ofusing a stationary AR process to describe downturns, one may argue that the downturnsin yt are generated by a negative bubble process (2), where β > 1 and bt < 0.

To see the time series property in the downturns more clearly, we plot the monthlydata between October 2006 and March 2009 in Figure 2. The plot seems to suggest analternative model of a quadratic trend to describe the downturns, rather than a stationaryAR model. A similar observation can be made for the data in the 1917 stock market crashperiod (1917M08-1918M04). These observations naturally raise two questions. First, ifthe time series data is actually generated by a downward quadratic trend model, is itpossible for right-tailed unit root tests to reject the null hypothesis of unit root in favor

3

09/06 02/07 07/07 12/07 05/08 10/08 03/10160

180

200

220

240

260

280

300

320

340

360

Figure 2: Monthly price-dividend ratio of S&P500 between October 2006 to March 2009.

of the explosive alternative? This question extends to the testing results of the explosivepositive bubble. That is, when an explosive positive bubble is detected by a right-tailedunit root test, is it possible that the data is actually generated from a model with anupward polynomial trend? Second, if the answer to the first question is yes, how tomodify the right-tailed unit root tests so that they are robust under deterministic trends.These two questions motivate us to study the ability of two right-tailed unit root testsbased on AR regressions to distinguish the explosive process from the trend-stationaryprocess, and to introduce robust unit root testing procedures.

In the paper, we first show that if the data is generated from a stationary process witha quadratic trend or a higher order trend (upward or downward), the LS estimate of βfrom regression (3) tends to be larger than 1 so that the right-tailed unit root tests (boththe t-test and the coeffi cient-based test) often reject the unit root null hypothesis in favorof the explosive alternative. We then propose to construct the right-tailed unit root testsbased on the following regression with an explicit requirement that k ≥ 1:

yt = α+ βyt−1 +∑k

i=1ψi∆yt−i + et. (4)

It is proved that, when the data generating process (DGP) is trend-stationary, the estimateβ in regression (4) converges to 1 at a rate faster than Op(n). As a result, both the t-testand the coeffi cient-based test based on β tend not to reject the null hypothesis of unitroot. Whereas, when the data is generated by a mildly explosive process, the test statisticsbased on β diverge to positive infinity, and hence, the right-tailed unit root tests rejectthe unit root null hypothesis in favor of the explosive alternative.

4

To see the difference in empirical implications of the tests based on (3) and on (4)with k = 1, we first apply the coeffi cient-based test recursively to the monthly S&P 500stock price index-dividend ratio from January 1871 to December 2010. It is found that thetest based on (4) with k = 1 detects the same bubbles as the test based on (3), includingfive positive bubble episodes and two negative bubble episodes. Since the right-tailedtest based on (4) with k = 1 can better distinguish the explosive process from the trend-stationary process, we can conclude that the seven episodes including the two negativebubble episodes are not likely due to polynomial trends.

In the second empirical study, we apply the coeffi cient-based test recursively to themonthly price-rent ratio of the US housing market (the Case-Shiller US National HomePrice Index divided by the rent of primary residence) for the period from January 1981to June 2017. The test based on (3) detects two positive bubble episodes (1986M05 to1990M04, and 1998M04 to 2008M01). Moreover, it also flags two collapsing periods asbubble episodes. However, the right-tailed coeffi cient-based test based on (4) with k = 1

detects only the two positive bubble periods, indicating that the data in the two collapsingperiods may be better fitted by models with downward polynomial trends. In addition, thetwo positive bubble episodes are estimated to end much earlier by our proposed methodthan those based on (3). The estimated termination dates of the two positive bubbles bythe proposed method match better with the turning points in the data.

The rest of the paper is organized as follows. Sections 2-4 introduce the stationaryprocesses with different kinds of trends, including the linear trend, the quadratic trend,and the cubic trend. For each trend-stationary process, the large sample theory of the LSestimator of the AR coeffi cient and the two unit root test statistics are developed, when(misspecified) AR models are fitted. The large sample results show that the right-tailedunit root tests based on the regression (3) cannot distinguish the explosive process from thetrend-stationary process. In Section 5 we show that the right-tailed unit root tests basedon the regression (4) with k ≥ 1 can successfully distinguish the mildly explosive processfrom the trend stationary process. Section 6 presents simulation evidence to support theasymptotic results. The proposed procedure is used to analyze two real time series inSection 7. Section 8 concludes. A brief summary and the proof of all theoretical resultsare included in the Appendix.

2 Linear Trend Model and Explosiveness

We first study the asymptotic performance of the right-tailed unit root tests based on theLS regressions of (4) with different values of k including k = 0, when the data is actuallygenerated from the linear trend model as

yt = δt+ ut, t = 1, 2, ..., n, (5)

where δ is a non-zero constant (positive or negative), ut = C(L)εt =∑∞

j=0 cjεt−j , c0 = 1,∑∞j=0 j|cj | < ∞, and {εt} is a sequence of independent and identically distributed (iid)

5

random variables with E (εt) = 0, V ar (εt) = σ2, and E(ε4t

)<∞. Let γ0 = V ar (ut) and

γ1 = Cov (ut, ut−1).We consider the three LS regression equations:

yt = βyt−1 + et, (6)

yt = α+ βyt−1 + et, (7)

yt = α+ βyt−1 +∑k

i=1ψi∆yt−i + et, (8)

where β,(α, β

), and

(α, β, ψ1, · · · , ψk

)are the conventional LS estimators. We define the

following regression t-statistics which are commonly used to test the unit root hypothesisagainst the explosive alternative:

tβ

=β − 1

se(β) , tβ =

β − 1

se(β) , and tβ =

β − 1

se(β) ,

where se(β), se

(β)and se

(β)are the standard errors of β, β and β, respectively. The

coeffi cient-based statistics are defined as n(β − 1

), n(β − 1

)and n

(β − 1

), respectively,

which can also be used to test the unit root hypothesis against explosiveness.Note that the process yt defined in (5) can be rewritten as

yt = δ + yt−1 + ∆ut, (9)

where ∆ut = ut − ut−1. Hence, the true DGP of yt is not covered by the regression (6)for any value of β due to the presence of an intercept in (9). Theorem 2.1 reports theasymptotic theory of the t-statistic and the coeffi cient-based statistic when the regression(6) is fitted to yt.

Theorem 2.1 When the LS regression (6) is fitted to the time series yt generated fromthe linear trend model defined in (5), we have, as n→∞,

(a) n(β − 1

)= 3/2 +Op

(n−1

);

(b) tβ/√n

p−→√

3 |δ| /√δ2 + 8 (γ0 − γ1).

Theorem 2.1 suggests that, although βp−→ 1 at the rate of n, the average value of β

is greater than one. Moreover, n(β − 1

)≈ 1.5 which is larger than 1.28, the 95% critical

value of the Dickey-Fuller (DF) distribution.2 In addition, the t-statistic tβdiverges to

+∞ with the speed of√n. Hence, when the data is generated from a linear trend model,

the coeffi cient-based statistic and, more so, the t-statistic in regression (6) tend to rejectthe unit root null hypothesis in favor of explosiveness, no matter what the sign of δ is.

2The DF distribution is given by∫ 1

0W (r) dW (r) /

∫ 1

0[W (r)]2 dr where W (r) is a standard Brownian

motion; see Hamilton (1994) for the textbook treatment and Table B.5 Case 1 in Hamilton (1994) for finitesample critical values and asymptotic critical values.

6

The regression equations (7) and (8), although misspecified, include the true DGP of

yt when particular values of the parameters are chosen:(α, β

)= (δ, 1) for regression

(7), and(α, β, ψ1, · · · , ψk

)= (δ, 1, 0, · · · , 0) for regression (8). Theorem 2.2 reports the

asymptotic theory for the t-statistic and the coeffi cient-based statistic when the regression(7) is fitted to the data yt.

Theorem 2.2 When the LS regression (7) is fitted to the time series yt generated fromthe linear trend model defined in (5), we have, as n→∞,

(a) n2(β − 1

)= −12 (γ0 − γ1) /δ2 + 6 (un + u1) /δ + op (1) ;

(b)√ntβ = −6(γ0−γ1)+3δ(un+u1)

|δ|√

6(γ0−γ1)+ op (1) .

Theorem 2.2 suggests that n(β − 1

)p−→ 0, indicating the null of unit root cannot be

rejected if a positive asymptotic critical value is used.3 Interestingly, it can be seen thatβ has a downward bias as

E(β)

= 1− 12 (γ0 − γ1) /(n2δ2) + o(n−2

).

Hence, the average value of tβ is expected to be negative. This is confirmed as

E(tβ

)=

−6 (γ0 − γ1)

|δ|√

6n (γ0 − γ1)+ o

(n−1/2

).

Note that tβ converges to zero, indicating the null of unit root cannot be rejected if apositive critical value is used.4 When the critical values for both tests are chosen to bepositive constants or to diverge to positive infinity at the same speed, since n

(β − 1

)converges in probability to zero faster than tβ, we expect that the coeffi cient-based testhas better asymptotic power than the t-test to distinguish the linear trend process fromthe explosive process.

Based on the large sample results reported in Theorem 2.3 below, similar conclusionscan be made when the regression (8) with k ≥ 1 is used to fit the time series yt generatedfrom the linear trend model (5). That is, both the coeffi cient-based test and the t-testtend not to reject the null hypothesis of unit root and the coeffi cient-based test has betterasymptotic power than the t-test to distinguish the linear trend process from the explosiveprocess.

Theorem 2.3 When the LS regression (8) with k ≥ 1 is fitted to the time series ytgenerated from the linear trend model defined in (5), we have, as n→∞,

(a) n2(β − 1

)= Op (1) ;

(b)√ntβ = Op (1) ,

where the form of the limiting distribution depends on the value of k in the regression.

3From Table B.5 in Hamilton (1994), the 97.5% and the 99% critical value are 0.41 and 1.04.4From Table B.6 in Hamilton (1994), the 97.5% and the 99% critical value are 0.23 and 0.60.

7

3 Quadratic Trend Model and Explosiveness

In this section, we study the asymptotic performance of the right-tailed unit root testsbased on the LS regressions of (6), (7) and (8), respectively, when the data {yt} is generatedfrom a quadratic trend model as

yt = δt2 + ut, t = 1, 2, ..., n, (10)

where δ is a positive or negative constant, {ut} is a weakly stationary process as definedin (5). An equivalent representation of yt is

yt = −δ + 2δt+ yt−1 + ∆ut. (11)

Due to the presence of both the intercept and the linear trend in (11), when theregression (6) is considered, no value of β can cover the true DGP of yt; when the regression

(7) is considered, no value of(α, β

)can cover the true DGP of yt. However, since ∆yt −

∆yt−1 = ∆2yt = 2δ + ∆2ut, the DGP of yt can be rewritten as

yt = 2δ + yt−1 + ∆yt−1 + ∆2ut. (12)

Therefore, the DGP of yt is covered by the regression (8) with parameters(α, β, ψ1, · · · , ψk

)=

(2δ, 1, 1, 0, · · · , 0).Theorems 3.1-3.3 report the asymptotic theory of the t-statistic and the coeffi cient-

based statistic when regressions (6)-(8) are fitted to yt, respectively.

Theorem 3.1 When the LS regression (6) is fitted to the data {yt} generated from thequadratic trend model defined in (10), we have, as n→∞,

(a) n(β − 1

)= 5/2 +Op

(n−1

);

(b) tβ/√n

p−→√

15.

Theorem 3.1 shows that, on average n(β − 1

)≈ 5/2 > 2.03, the 99% critical value of

the DF distribution. In addition, the t-statistic diverges to +∞ at the speed of√n. Note

that the leading terms both in β − 1 and tβare independent of δ. Therefore, no matter

what the sign of δ has, the coeffi cient-based test, and more so, the t-test in regression (6)tend to reject the unit root null hypothesis in favor of explosiveness.

Theorem 3.2 When the LS regression (7) is fitted to the data {yt} generated from thequadratic trend model defined in (10), we have, as n→∞,

(a) n(β − 1

)= 15/8 +Op

(n−1

);

(b) tβ/√n

p−→√

15.

When the regression (7) is considered, the 99% asymptotic critical values of the DFcoeffi cient-based test and the DF t-test are 1.04 and 0.6, respectively. However, the large

8

sample results in Theorem 3.2 show that n(β − 1

)≈ 1.875 > 1.04, and tβ ≈

√15n →

+∞. As a result, for any value of δ (> 0 or < 0), both the coeffi cient-based test andthe t-test in the regression (7) tend to reject the unit root null hypothesis in favor ofexplosiveness. It means that these two tests, especially the t-test, have no power todistinguish the quadratic trend process from the explosive process under the LS regression(7). Since the regression (7) is the one that is commonly used in the literature on testingand dating bubbles, this finding has important empirical implications.

Theorem 3.3 When the LS regression (8) with k ≥ 1 is fitted to the data {yt} generatedfrom the quadratic trend model defined in (10), we have, as n→∞,

(a) n3(β − 1

)= Op (1) ;

(b)√ntβ = Op (1),

where the form of the limiting distribution depends on the value of k in the regression (8).

The large sample results reported in Theorem 3.3 show that n(β − 1

) p−→ 0 and

tβp−→ 0, which suggest that both the coeffi cient-based test and the t-test in regression (8)

with k ≥ 1 tend not to reject the unit root null hypothesis if the critical values for bothtests are chosen to be positive constants or to diverge to positive infinity. Since n

(β − 1

)converges in probability to zero faster than tβ, we expect that the coeffi cient-based testhas better asymptotic power than the t-test to distinguish the quadratic trend processfrom the explosive process under the LS regression (8) with k ≥ 1.

4 Cubic Trend Model and Explosiveness

In this section, we assume the data {yt} is generated from a cubic trend model as

yt = δt3 + ut, t = 1, 2, ..., n, (13)

where δ is a non-zero constant and {ut} is a weakly stationary process defined as in (6).An equivalent representation of the DGP of yt is

yt = 3δt2 − 3δt+ δ + yt−1 + ∆ut. (14)

Due to the presence of the intercept, the linear trend, and the quadratic trend in (14), theregression (6) does not cover the true DGP of yt for any value of β, and the regression (7)

does not cover the true DGP of yt for any value of(α, β

).

Note that ∆2yt = 6δ (t− 1) + ∆2ut, which leads to

yt = −6δ + yt−1 + ∆yt−1 + 6δt+ ∆2ut.

Together with the result of ∆2yt−1 = 6δ (t− 2) + ∆2ut−1, an alternative representation ofthe DGP of yt can be obtained as

yt = 6δ + yt−1 + 2∆yt−1 −∆yt−2 + ∆3ut. (15)

9

Hence, the true DGP is covered by the regression (8) with k ≥ 2 and(α, β, ψ1, ψ2, · · · , ψk

)=

(6δ, 1, 2,−1, 0, · · · , 0). Since ∆yt−1 = 3δt2 +Op (t) and ∆yt−2 = 3δt2 +Op (t), the regres-sion (8) with k ≥ 2 faces with the problem of asymptotic perfect collinearity. However, aswe prove in the Appendix, this problem plays no effect on the large sample property of β.

Theorems 4.1-4.3 report the asymptotic theory of the t-statistic and the coeffi cient-based statistic when regressions of (6)-(8) are fitted, respectively, to the data {yt} gener-ated from the cubic trend model (13).

Theorem 4.1 When the LS regression (6) is fitted to the data {yt} generated from thecubic trend model (13), we have, as n→∞,

(a) n(β − 1

)= 7/2 +Op

(n−1

);

(b) tβ/√n

p−→√

35.

Theorem 4.1 shows that, n(β − 1

)≈ 7/2 > 2.03, the 99% critical value of the DF

distribution. Moreover, the t-statistic tβdiverges towards +∞ at the speed of

√n. There-

fore, no matter what sign δ has, the coeffi cient-based test and, more so, the t-test in theregression (6) tend to reject the unit root null hypothesis in favor of explosiveness.

Theorem 4.2 When the LS regression (7) is fitted to the data {yt} generated from thecubic trend model (13), we have, as n→∞,

(a) n(β − 1

)= 28/9 +Op

(n−1

);

(b) tβ/√n

p−→√

35.

From Theorem 4.2, it can be seen that tβ ≈√

35n→ +∞ as n→∞, and n(β − 1

)≈

28/9 > 1.04, the 99% critical value of the DF distribution. Thus, for any non-zero valueof δ (> 0 or < 0), both the coeffi cient-based test, and more so, the t-test in regression (7)tend to reject the unit root null hypothesis in favor of explosiveness.

Theorem 4.3 When the LS regression (8) is fitted to the data {yt} generated from thecubic trend model (13), we have, as n→∞,

(a) when k = 1,

n2(β − 1

)= −8.4 +Op

(n−1

), and tβ/

√n = −

√21/2 +Op

(n−1

);

(b) when k > 1,

n4(β − 1

)= Op (1) , and

√ntβ = Op (1) .

where the limits of β and tβ in (b) depends on the value of k in the regression (8).

10

From (15), it is known that the true DGP of yt is covered by the regression (8) withk > 1. Part (b) of Theorem 4.3 shows that n

(β − 1

) p−→ 0 and tβp−→ 0. Hence, both the

coeffi cient-based test and the t-test from the regression (8) with k > 1 tend not to rejectthe unit root null hypothesis if the critical values for both tests are chosen to be positiveconstants or to diverge to positive infinity. Since n

(β − 1

)converges in probability to

zero much faster than tβ, we expect that the coeffi cient-based test has better asymptoticpower than the t-test to distinguish the cubic trend process from the explosive process.

Also revealed by the representation in (15) is that the regression (8) with k = 1 doesnot cover the true DGP of yt. Part (a) of Theorem 4.3 shows that, although β convergesto 1 less quickly than it does in the regression (8) with k > 1, it still goes to 1 fast enoughto ensure n

(β − 1

) p−→ 0. Moreover, the leading term of n(β − 1

)is negative. Therefore,

the coeffi cient-based test tends not to reject the unit root null hypothesis if a positivecritical value is used. It is also shown that tβ diverges to negative infinity at the speed ofn1/2, which implies that the t-test tends not to reject the unit root null hypothesis.

It is not diffi cult to extend the theory to the case where the true DGP has a deter-ministic trend with an order higher than 3. In particular, if the LS regression (8) withk = 1 is applied to fit the data, it can still be shown that n

(β − 1

) p−→ 0 and tβp−→ −∞.

Hence, both the t-test and the coeffi cient-based test tend not to reject the unit root nullhypothesis.

5 A Proposed Regression

As we have proved in earlier sections, if the true DGP is a trend-stationary process, theright-tailed unit root tests based on the regression model (8) with k ≥ 1 tend not to findthe evidence of explosiveness and hence can distinguish a bubble behavior from a trend-stationary behavior. This finding motivates us to propose the following procedure to testthe unit root null hypothesis against explosiveness:

yt = α+ βyt−1 +

k∑i=1

ψi∆yt−i + et with k ≥ 1, (16)

where the minimum value that k can take is set to be 1, namely at least one term involvingthe lagged ∆yt is included in the regression. As shown in the following Theorem 5.1,the right-tailed unit root tests (the coeffi cient-based test and the t-test) based on the LSregression of (16) are able to distinguish the explosive AR process from the trend-stationaryprocess.

Theorem 5.1 Let W (r) denote a standard Brownian motion, and ut =∑∞

j=0 cjεt−j be

the weakly stationary process as defined in (5) with εtiid∼ N(0, σ2).

(a) When the DGP is yt = yt−1 + εt, the LS regression (16) with k ≥ 1 leads to

n(β − 1

)⇒∫ 1

0 W (r) dW (r)−W (1)∫ 1

0 W (r) dr∫ 10 [W (r)]2 dr −

(∫ 10 W (r) dr

)2 , (17)

11

and

tβ =β − 1

se(β) ⇒ ∫ 1

0 W (r) dW (r)−W (1)∫ 1

0 W (r) dr{∫ 10 [W (r)]2 dr −

(∫ 10 W (r) dr

)2}1/2

, (18)

as n→∞;(b) When the DGP is an ARIMA(p, 1, 0) process with p ≥ 1, the statistics n

(β − 1

)and

tβ based on the LS regression (16) with k ≥ p have the same limiting distributions as thosegiven in (a), respectively;(c) When the DGP is yt = δtm + ut with m = 1 or 2 or 3,..., the statistics n

(β − 1

)and

tβ based the LS regression (16) with k ≥ 1 have the limits as

n(β − 1

) p−→ 0 and tβp−→ 0 or −∞;

(d) When the DGP is mildly explosive, i.e., yt = ρnyt−1 + ut with ρn = 1 + c/nθ, c > 0,and θ ∈ (0, 1), the statistics n

(β − 1

)and tβ based on the LS regression (16) with k ≥ 1

have the limits asn(β − 1

)→ +∞ and tβ → +∞.

The results reported in Part (c) of Theorem 5.1 show that, if the data is generatedfrom a trend-stationary model, regardless of the direction of the trend, n

(β − 1

) p−→ 0

and tβp−→ 0 or −∞. Hence, the right-tailed unit root tests from the regression (16) do

not reject the unit root null hypothesis when the critical values for both tests are chosento be positive constants or to diverge to positive infinity. Whereas, as reported in Part (d)of Theorem 5.1, if the data is generated from a mildly explosive process, n

(β − 1

)→ +∞

and tβ → +∞. Hence, both the coeffi cient-based test and the t-test should reject the unitroot null hypothesis in favor of explosive alternative. As a result, both the two right-tailedunit root tests based on the regression (16) with k ≥ 1 can distinguish the mildly explosiveprocess from the trend-stationary process.

6 Simulation Studies

6.1 Linear trend model

Tables 1-3 report some simulation results based on 10,000 replications for the LS regres-sions (6), (7), and (8) with k = 1, respectively. For each regression, the true DGP is

the linear trend model as in (5) with δ = −2 or −4, utiid∼ N(0, 1), and n = 20 or 50.

Unless explicitly stated, we always report the mean and variance of the LS estimator ofthe AR(1) coeffi cient, the minimum, mean and maximum of the coeffi cient-based statis-tic (i.e. c-stat), and the minimum, mean and maximum of the t-statistic (i.e. t-stat).Also reported are the 99% finite sample critical values of the c-stat and t-stat which aresimulated under the random walk null hypothesis.

Table 1 reports the simulation results for the regression (6), an AR(1) regression with-out intercept. Some conclusions can be made from Table 1. First, the average values of β

12

Table 1: Statistical results for the LS regression (6) when the data is generated from alinear trend model as in (5). The 99% finite sample critical values of the test statistics aresimulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 1.0749 1.0764 1.0300 1.0302

variance of β 1.76e-05 4.32e-06 3.90e-07 9.67e-08minimum of c-stat 1.1426 1.3092 1.3473 1.4195mean of c-stat 1.4226 1.4517 1.4696 1.4810maximum of c-stat 1.7351 1.6024 1.5794 1.537899% CV of c-stat 2.37 2.37 2.16 2.16minimum of t-stat 2.1047 4.2141 4.7689 8.0217mean of t-stat 4.3897 6.3537 7.0046 10.0048maximum of t-stat 8.4469 8.7309 9.3513 11.707799% CV of t-stat 2.24 2.24 2.08 2.08

in all four cases are greater than 1, although getting closer to 1 as n increases. Second, theaverage values of the c-stat in all four cases are around 1.5, which is the value suggestedby the asymptotic theory given in Theorem 2.1. In addition, except for the case whereδ = −2 and n = 20, the minimum values of the c-stat are all larger than 1.28 which is the95% asymptotic critical value of the DF distribution. Hence, if this critical value is used,the unit root null hypothesis will always be rejected in favor of the explosive alternative.However, when the 99% finite sample critical values are used, the unit root null hypothesiswill not be rejected. Third, except for the case with δ = −2 and n = 20 in which theminimum value of the t-stat is slightly smaller than the corresponding 99% finite samplecritical value, the minimum values of the t-stat in other cases are all larger than the corre-sponding 99% finite sample critical values, indicating that the unit root null hypothesis isalways rejected in favor of the explosive alternative. Fourth, the simulated average valuesof the t-stat are consistent with that suggested by the asymptotic theory given in Theorem2.1. For example, according to the asymptotic theory, when ut

iid∼ N(0, 1) and δ = −2, weshould have t

β/√n

p→ 2√

3/√

4 + 8 = 1, suggesting that tβ≈√n. Hence, t

β≈ 4.47 when

n = 20, and tβ≈ 7.07 when n = 50. These values are very close to the simulated average

values of the t-stat reported in Table 1.Table 2 reports the simulation results for the regression (7), an AR(1) regression with

an intercept. Some conclusions can be made from Table 2. First, β converges to one veryquickly. Second, β has a small downward bias. This explains why the average values ofthe c-stat and the t-stat are negative. Third, the maximum values of the c-stat in all casesare smaller than the corresponding 99% finite-sample critical values, indicating that theunit root null hypothesis will not be rejected in favor of the explosive alternative. Fourth,although the maximum values of the t-stat in all cases are larger than the respective99% finite-sample critical values, their average values are much smaller than these critical

13

Table 2: Statistical results for the LS regression (7) when the data is generated from alinear trend model as in (5). The 99% finite sample critical values of the test statistics aresimulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 0.9919 0.9980 0.9988 0.9997

variance of β 1.44e-04 3.47e-05 3.24e-06 7.94e-07minimum of c-stat -1.0523 -0.4523 -0.4051 -0.1846mean of c-stat -0.1545 -0.0389 -0.0610 -0.0153maximum of c-stat 0.6914 0.3951 0.2610 0.142599% CV of c-stat 1.40 1.40 1.22 1.22minimum of t-stat -2.0613 -1.8580 -1.2086 -1.1304mean of t-stat -0.2777 -0.1397 -0.1737 -0.0868maximum of t-stat 1.5702 1.6110 0.8810 0.964199% CV of t-stat 0.80 0.80 0.66 0.66

values. Hence, the t-test will not reject the unit root null hypothesis most of the time.However, it can be seen that the coeffi cient-based test is more powerful than the t-test todistinguish the linear trend process from the explosive process in finite samples. Fifth, thesimulation results are consistent with what suggested by the asymptotic theory given in 2.2.For example, when ut

iid∼ N(0, 1), n = 50 and δ = −4, the asymptotic theory suggests that

E(β)

= 1− 12/(n2δ2

)+ o

(n−2

)≈ 0.9997 and E

(tβ

)= −

√6

|δ|√n

+ o (1/√n) ≈ −0.0866.

These values are nearly identical to what we obtained in simulations.Table 3 reports the simulation results for the regression (8) with k = 1. Conclusions

similar as those obtained from Table 2 can be made from Table 3. The most importantone is that β converges to 1 very quickly making the c-stat taking values around 0 witha small variation. As a result, the right-tailed unit root test based on the c-stat cannotreject the unit root null hypothesis, and hence, has power to distinguish the linear trendprocess from the explosive process.

6.2 Quadratic trend model

Tables 4-6 report some simulation results based on 10,000 replications for the LS regres-sions (6), (7), and (8) with k = 1, respectively. For each regression, the true DGP is the

quadratic trend model as in (10) with δ = −2 or −4, utiid∼ N(0, 1), and n = 20 or 50.

Table 4 reports the simulation results for the regression (6), an AR(1) model withoutintercept. Some conclusions can be made from Table 4. First, the average value of β isgreater than 1 in all four cases, although gets closer to 1 in the cases with larger n. Theaverage value of β is very close to 1 + 2.5/n, a value predicted by our asymptotic theorygiven in Theorem 3.1. Second, the minimum values of the c-stat in all cases are larger thanthe respective 99% finite sample critical values. Hence, the coeffi cient-based test alwaysindicates explosiveness in unit root testing. Third, consistent with the asymptotic theory

14

Table 3: Statistical results for the LS regression (8) with k = 1 when the data is generatedfrom a linear trend model as in (5). The 99% finite sample critical values of the teststatistics are simulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 0.9955 0.9988 0.9994 0.9998

variance of β 2.15e-04 5.35e-05 4.23e-06 1.05e-06minimum of c-stat -0.9786 -0.4748 -0.4051 -0.1939mean of c-stat -0.0815 -0.0207 -0.0308 -0.0076maximum of c-stat 1.0156 0.5005 0.3478 0.181299% CV of c-stat 2.29 2.29 1.50 1.50minimum of t-stat -1.9943 -1.9186 -1.2923 -1.2605mean of t-stat -0.1566 -0.0804 -0.0983 -0.0489maximum of t-stat 2.1857 2.2014 1.3951 1.432399% CV of t-stat 0.92 0.92 0.72 0.72

Table 4: Statistical results for the LS regression (6) when the data is generated from aquadratic trend model as in (10). The 99% finite sample critical values of the test statisticsare simulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 1.1327 1.1327 1.0512 1.0512

variance of β 1.32e-07 3.30e-08 4.46e-10 1.11e-10minimum of c-stat 2.4966 2.5091 2.5042 2.5063mean of c-stat 2.5212 2.5214 2.5084 2.5084maximum of c-stat 2.5475 2.5345 2.5124 2.510499% CV of c-stat 2.37 2.37 2.16 2.16minimum of t-stat 15.0876 15.5004 26.2124 26.3187mean of t-stat 15.7351 15.8186 26.3792 26.4052maximum of t-stat 16.4100 16.1522 26.5231 26.479499% CV of t-stat 2.24 2.24 2.08 2.08

15

Table 5: Statistical results for the LS regression (7) when the data is generated from aquadratic trend model as in (10). The 99% finite sample critical values of the test statisticsare simulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 1.0944 1.0944 1.0376 1.0376

variance of β 3.70e-07 9.26e-08 1.27e-09 3.18e-10minimum of c-stat 1.7491 1.7712 1.8353 1.8388mean of c-stat 1.7929 1.7932 1.8422 1.8422maximum of c-stat 1.8367 1.8151 1.8494 1.845899% CV of c-stat 1.40 1.40 1.22 1.22minimum of t-stat 14.8184 16.6229 26.9877 27.3744mean of t-stat 17.2173 17.6987 27.5395 27.6500maximum of t-stat 19.7671 19.0951 28.0039 27.877199% CV of t-stat 0.80 0.80 0.66 0.66

given in Theorem 3.1, in all cases, the t-stat takes values around√

15n with a very smallvariation. In addition, the minimum values of the t-stat in all cases are larger than therespective 99% finite sample critical values. Hence, the t-test always rejects the unit rootnull hypothesis in favor of explosive alternative. All these findings corroborate the largesample theory given in Theorem 3.1.

Table 5 reports the simulation results for the regression (7), an AR(1) model with anintercept. Similar conclusions to those from Table 4 can be made from Table 5. First,the average value of β is greater than 1 in all four cases, although it gets closer to 1 asn increases. Second, consistent with the asymptotic theory given in Theorem 3.2, thec-stat takes values around 15/8 ≈ 1.8750 with a very small variation so that its minimumvalues in all cases are larger than the respective 99% finite sample critical values. Hence,the coeffi cient-based test always suggests explosiveness in unit root testing. Third, assuggested by the asymptotic theory, the t-stat in every case takes values around

√15n with

a small variation. As√

15n is much larger than the respective 99% finite sample criticalvalues, the t-test always rejects the unit root null hypothesis in favor of the explosivealternative. These findings from Table 5 have important empirical implications as theimplementation of the right-tailed unit root testing in the literature has been based on theLS regression (7). When the data is generated from a quadratic trend model, our findingssuggest that the unit root tests based on the regression (7) always suggest explosiveness.

Table 6 reports the simulation results for the regression (8) with k = 1. Severalconclusions are made from Table 6. First, β converges to 1 very quickly so that the c-stattakes values around 0 with small variations in all cases. The maximum values of the c-statin all cases are smaller than the respective 99% finite sample critical values. Hence, theright-tailed coeffi cient-based test from the regression (8) with k = 1 does not reject thenull hypothesis of unit root. Second, the t-stat slowly goes to zero when n increases,

16

Table 6: Statistical results for the LS regression (8) with k = 1 when the data is generatedfrom a quadratic trend model as in (10). The 99% finite sample critical values of the teststatistics are simulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 1.0113 1.0030 1.0006 1.0002

variance of β 4.22e-05 7.98e-06 1.01e-07 2.030e-08minimum of c-stat -0.1857 -0.1301 -0.0273 -0.0189mean of c-stat 0.2042 0.0548 0.0299 0.0075maximum of c-stat 0.8005 0.3052 0.0933 0.037399% CV of c-stat 2.29 2.29 1.50 1.50minimum of t-stat -1.1180 -1.3639 -0.6453 -0.8976mean of t-stat 0.9370 0.4745 0.5872 0.2936maximum of t-stat 2.6522 2.1970 1.5824 1.286299% CV of t-stat 0.92 0.92 0.72 0.72

as suggested by the asymptotic theory given in Theorem 3.3. In most of the cases, theaverage value of the t-stat is smaller than the corresponding 99% critical value, indicatingthat the t-test has a good chance not to reject the unit root hypothesis. However, theprobability for the t-test to suggest explosiveness is non-negligible in finite samples becausethe maximum values of the t-stat in most of the cases are considerably larger than therespective 99% finite sample critical values.

To further understand the behavior of the t-stat, Table 7 reports extra simulationresults of the t-stat when the data is generated from the quadratic trend model as in (10)with δ = −4, n = 100 or 250. It can be seen that, as n gets larger, the probability forthe t-test to find explosiveness becomes smaller. And in the case where δ = −4 and n =

250, the maximum value of the t-stat becomes smaller than the corresponding 99% criticalvalue, and hence, the t-test does not find explosiveness any more. Comparing the t-statwith the c-stat, we conclude that the coeffi cient-based test is more able to distinguish thequadratic trend process from the explosive process in finite samples.5

6.3 Cubic trend model

Tables 8-10 report some simulation results based on 10,000 replications for the LS regres-sions (6), (7), and (8) with k = 1 and k = 2, respectively. For each regression, the true

DGP is the cubic trend model as in (13) with δ = −0.02 or −0.04, utiid∼ N(0, 1), and

n = 20 or 50.Table 8 reports the simulation results for the regression (6), an AR(1) model without

5This conclusion is consistent with that suggested by the large sample theory given in Theorem 3.3,where it is shown that the t-stat = Op (1/

√n) and the c-stat = Op

(1/n2

). In fact, when δ = −2, the ratio

of the average value of the t-stat for n = 20 and for n = 50 is 0.9370/0.5872 = 1.5957 which is very closeto√50/20 = 1.5811, reinforcing the result derived in Theorem 3.3 about the

√n-convergence of tβ .

17

Table 7: Further statistical results of the t-statistic from the regression (8) with k = 1when data is generated from the quadratic trend model (10).

δ -4 -4n 100 250minimum of t-stat -0.4716 -0.2782mean of t-stat 0.2079 0.1298maximum of t-stat 0.9289 0.571799% CV of t-stat 0.66 0.63

Table 8: Statistical results for the LS regression (6) when the data is generated from acubic trend model as in (13). The 99% finite sample critical values of the test statisticsare simulated under the random walk null hypothesis.

δ -0.02 -0.04 -0.02 -0.04n 20 20 50 50

mean of β 1.1907 1.1909 1.0724 1.0725

variance of β 7.18e-06 1.79e-06 3.64e-09 9.10e-10minimum of c-stat 3.4441 3.5376 3.5379 3.5440mean of c-stat 3.6227 3.6278 3.5500 3.5501maximum of c-stat 3.8245 3.7275 3.5614 3.555899% CV of c-stat 2.37 2.37 2.16 2.16minimum of t-stat 12.3593 18.1769 38.8708 39.5090mean of t-stat 18.8805 22.1239 39.6705 39.9027maximum of t-stat 25.1421 25.6325 40.3300 40.219899% CV of t-stat 2.24 2.24 2.08 2.08

18

Table 9: Statistical results for the LS regression (7) when the data is generated from acubic trend model as in (13). The 99% finite sample critical values of the test statisticsare simulated under the random walk null hypothesis.

δ -0.02 -0.04 -0.02 -0.04n 20 20 50 50

mean of β 1.1654 1.1659 1.0639 1.0639

variance of β 1.41e-05 3.52e-06 7.26e-09 1.81e-09minimum of c-stat 2.8777 3.0176 3.1126 3.1208mean of c-stat 3.1428 3.1519 3.1288 3.1289maximum of c-stat 3.4192 3.2883 3.1458 3.137499% CV of c-stat 1.40 1.40 1.22 1.22minimum of t-stat 8.7165 14.6320 38.2528 39.6795mean of t-stat 15.8692 21.1515 39.8576 40.4008maximum of t-stat 27.4502 27.7903 41.1378 41.025699% CV of t-stat 0.80 0.80 0.66 0.66

intercept. Some conclusions can be made from Table 8. First, the average value of β isgreater than 1 in all four cases, and gets closer to 1 as n increases. Second, in all cases,the c-stat takes values around 3.5 with a very small variation, a value predicted by ourasymptotic theory given in Theorem 4.1. Moreover, the minimum values of the c-statin all cases are larger than the respective 99% finite sample critical values. Hence, thecoeffi cient-based test always indicates explosiveness in unit root testing. Third, consistentwith the asymptotic theory given in Theorem 4.1, in all cases, the t-stat takes valuesaround

√35n with a small variation. Consequently, the minimum values of the t-stat in

all cases are larger than the respective 99% finite sample critical values. Hence, the t-testalways rejects the unit root null hypothesis in favor of explosive alternative. All thesefindings corroborate the large sample theory given in Theorem 4.1.

Table 9 reports the simulation results for the regression (7), an AR(1) model with anintercept. Similar conclusions to those from Table 7 can be made from Table 9. First, theaverage value of β is greater than 1 in all four cases, and get closer to 1 as n increases.Second, consistent with the asymptotic theory given in Theorem 4.2, the c-stat takes valuesaround 28/9 ≈ 3.1111 with a very small variation so that its minimum values in all casesare larger than the respective 99% finite sample critical values. Hence, the coeffi cient-based test always suggests explosiveness in unit root testing. Third, as suggested bythe asymptotic theory, the t-stat in every case takes values around

√35n with a small

variation. As√

35n is much larger than the respective 99% finite sample critical values,the t-test also always find explosiveness in unit root testing. All these findings corroboratethe large sample theory given in Theorem 4.2.

Table 10 reports the simulation results for the regression (8) with k = 1. Some conclu-sions different with those from Tables 7-8 can be made from Table 9. First, as predictedby Part (1) of Theorem 4.3, β converges to 1 very quickly. When n and the absolute value

19

Table 10: Statistical results for the LS regression (8) with k = 1 when the data is generatedfrom a cubic trend model as in (13). The 99% finite sample critical values of the teststatistics are simulated under the random walk null hypothesis.

δ -0.02 -0.04 -0.02 -0.04 -0.02 -0.04n 20 20 50 50 250 250

mean of β 1.1491 1.0626 1.0005 0.9976 0.9999 0.9999

variance of β 0.0016 8.83e-04 1.67e-06 2.43e-07 1.39e-12 3.44e-13minimum of c-stat -0.4477 -0.6767 -0.1719 -0.1955 -0.0341 -0.0338mean of c-stat 2.6846 1.1275 0.0222 -0.1141 -0.0331 -0.0333maximum of c-stat 5.2222 3.3061 0.2875 -0.0136 -0.0320 -0.032799% CV of c-stat 2.29 2.29 1.50 1.50 1.02 1.02minimum of t-stat -0.6116 -1.5677 -1.5055 -3.3262 -14.6752 -24.1792mean of t-stat 3.2143 1.5686 0.1023 -1.4358 -11.6925 -20.4927maximum of t-stat 7.2216 4.0850 1.3016 -0.1418 -9.4362 -17.407999% CV of t-stat 0.92 0.92 0.72 0.72 0.63 0.63

of δ are reasonably large, β has downward bias. Second, the c-stat gets closes to zero as nincreases whereas the t-stat diverges as n increases. When n = 50, the maximum valuesof the c-stat are smaller than the respective 99% finite sample critical values, indicatingthat the coeffi cient-based test will not suggest explosiveness in unit root testing regardlessof δ = −0.02 or −0.04. Third, in the cases where n = 50 and δ = −0.04, the maximumvalues of the t-stat are smaller than the respective 99% finite sample critical values, indi-cating that the t-test will not reject unit root null hypothesis. However, when n = 20 orwhen n = 50 and δ = −0.02, the t-test has to reject the null hypothesis of unit root infavor of explosiveness for some replications. Hence, the coeffi cient-based test is more ableto distinguish the cubic trend process from the explosive process than the t-test in finitesamples.

To further understand the behavior of the two statistics under the regression (8) withk = 1, we have done extra calculations for n = 250 and report the results in the last twocolumns of Table 10. The average value of β becomes smaller than one. Furthermore, theaverage values of the two statistics are close to what Part (1) of Theorem 4.3 predicts. Forexample, the average value of the coeffi cient-based statistics is close to −8.4/n = −0.0336.The average value of the t-statistics is closer to −

√21× n/2 ≈ −36. Both tests cannot

reject the null hypothesis of unit root in all cases.Table 11 reports the simulation results for the regression (8) with k = 2. It can be

seen that β− 1, the c-stat, and the t-stat converge to zero very fast. As a result, both thecoeffi cient-based test and the t-test will not find evidence for explosiveness. These findingscorroborate the large sample theory reported in Part (2) of Theorem 4.3.

20

Table 11: Statistical results for the LS regression (8) with k = 2 when the data isgenerated from a cubic trend model as in (13). The 99% finite sample critical values ofthe test statistics are simulated under the random walk null hypothesis.

δ -2 -4 -2 -4n 20 20 50 50

mean of β 0.9914 0.9973 0.9998 1.0000

variance of β 1.41e-05 3.24e-06 4.61e-09 7.79e-10minimum of c-stat -0.4015 -0.1927 -0.0225 -0.0071mean of c-stat -0.1464 -0.0465 -0.0075 -0.0019maximum of c-stat 0.0610 0.0626 0.0024 -0.003199% CV of c-stat 3.7344 3.7344 1.9928 1.9928minimum of t-stat -4.0344 -2.7740 —2.1171 -1.6387mean of t-stat -1.6916 -0.8830 -0.9371 -0.4696maximum of t-stat 1.0553 1.3609 0.4687 0.751899% CV of t-stat 1.1982 1.1982 0.8499 0.8499

Table 12: The percentiles of the coeffi cient-based statistic and the t-statistic based on theLS regression (16) with k = 1 when the true DGP is yt = yt−1 + ut.

n 90 95 99c-stat t-stat c-stat t-stat c-stat t-stat

20 -0.5299 -0.2422 0.3415 0.1435 2.2865 0.924050 -0.7389 -0.3743 0.0342 0.0175 1.4955 0.7208100 -0.8221 -0.4180 -0.0520 -0.0337 1.2574 0.6592250 -0.8144 -0.4058 -0.0957 -0.0556 1.0203 0.6283500 -0.8222 -0.4128 -0.1448 -0.0844 1.0641 0.5921∞ -0.85 -0.44 -0.13 -0.07 1.04 0.60

6.4 Proposed regression model

In this section, we fit simulated data to the LS regression (16) with k = 1. First, weobtain the critical values of the coeffi cient-based test and the t-test when the true DGPis yt = yt−1 + ut with y0 = 10 and ut

iid∼ N(0, 1). Both the c-stat and the t-stat arecalculated for 10,000 replications and we report the 90%, 95% and 99% critical values forn = 20, 50, 100, 250, 500 in Table 11. Also reported are the 90%, 95% and 99% criticalvalues of the asymptotic distributions obtained from Table B.5 and Table B.6 in Hamilton(1994). These critical values can be used to test the null hypothesis of unit root against theexplosive alternative in the proposed regression (16) with k = 1. The asymptotic criticalvalues can be used to test the null hypothesis of unit root against the explosive alternativein the proposed regression (16) with any k. When the test statistics take values largerthan the corresponding critical values, the evidence of explosiveness is found.

Second, in Table 12 we report the proportions of replications, out of 10,000 replications,where the coeffi cient-based test and the t-test reject the unit root null hypothesis in favor

21

Table 13: Proportion of replications for unit root tests to reject the null hypothesis of unitroot in favor of explosiveness based on the proposed regression

linear trend quadratic trend explosive (β = 1.03) explosive (β = 1.05)n c-stat t-stat c-stat t-stat c-stat t-stat c-stat t-stat50 0 0.007 0 0.043 0.268 0.700 0.984 0.995100 0 0.001 0 0.006 0.987 0.987 1.000 1.000250 0 0 0 0.001 1.000 1.000 1.000 1.000

of explosiveness when the true DGP is yt = −4t+ut, or yt = −4t2 +ut, or yt = βyt−1 +utwith y0 = 10 and β = 1.03 or β = 1.05, respectively. The two values of β for theexplosive process are empirically reasonable. In all DGPs, ut

iid∼ N(0, 1) and three differentsample sizes are used, i.e., n = 50, 100, 250. When the c-stat (t-stat) is larger thanthe corresponding 99% finite sample critical value, the unit root test finds evidence ofexplosiveness in the simulated data. The overall conclusion from this simulation studyis that our proposed procedure can effectively distinguish the trend-stationary processesfrom the explosive process. In particular, when data is generated from the trend-stationarymodels, in no replication the coeffi cient-based test based on the proposed regression findsevidence of explosiveness in all six cases considered. In a small number of replications,the t-test based on the proposed regression finds the evidence of explosiveness in unit roottesting. The proportion becomes smaller when the sample size increases. When data isgenerated from an explosive process with a stronger explosive behavior (β = 1.05), thetwo tests almost always find evidence of explosiveness. When the explosive behavior is notso strong (β = 1.03) and the sample size is small (n = 50), the two statistics, especiallythe coeffi cient-based statistic, has diffi culty in rejecting the unit root hypothesis. Whenn = 100 or 200, the two unit root tests almost always find the evidence of explosivenessregardless of β = 1.03 or 1.05.

From the above simulations, it is clear that there is a trade-off between the coeffi cient-based test and the t-test. While the c-statistic is more robust against trend-stationarityin data, it is less powerful in identifying a mildly explosive behavior in small samples. Forconservative users whose primary concern is on the robustness property of the right-tailedunit root testing against the trend stationary behavior, our recommendation is to use thecoeffi cient-based test.

7 Empirical Studies

The empirical usefulness of the right-tailed unit root tests has been made clear in PSY(2015a) for testing for the presence of bubbles and for dating each bubble. The PSYprocedure relies on repeated calculations of the t-statistic in autoregression in a recursivemanner where the end point r2 (fraction) of each sample takes a value between r0 to 1

and the starting point r1 (fraction) of the sample takes a value between 0 to r2 − r0 withr0 (fraction) being the smallest sample window. So bnr0c, the integer part of nr0, is the

22

minimum window size in the calculations. PSY (2015a) proposed the GSADF statistic tobe the largest t-statistic in the double recursion over all possible combinations of r1 andr2, namely

GSADF (r0) = supr2∈[r0,1],r1∈[0,r2−r0]

{tr2r1}, (19)

where tr2r1 is the t-statistic based on the sample from r1 to r2. PSY (2015a) derived theasymptotic distribution of GSADF (r0) when the null hypothesis is a unit root process,from which the right-tailed critical values can be obtained. The intuition why the test isreasonable is that if there is a subsample of data corresponding to an explosive bubbleperiod, the t-statistic calculated from this subsample should take a large value. Theproposed test is done in a recursive way to find such a subsample by seeking the largestvalue of the t-statistic in different subsamples. Clearly, the GSADF test can also be donebased on the coeffi cient-based statistics, and the corresponding limiting distribution underthe unit root null hypothesis can be easily obtained based on the results given in Part (a)of Theorem 5.1.

After the presence of bubbles has been detected, one can estimate the origination dateand the termination date of each bubble by

re = infr2∈[r0,1]

{r2 : BSADFr2 (r0) > cvr0} , (20)

rf = infr2∈[re,1]

{r2 : BSADFr2 (r0) < cvr0} , (21)

whereBSADFr2 (r0) = sup

r1∈[0,r2−r0]

{tr2r1}, (22)

and cvr0 is the critical value of the sup t-statistic. Phillips, Wu and Yu (2011) developed the

asymptotic distribution of the sup t-statistic as supr∈[r0,1]

∫ r0 WrdW/

(∫ r0 W

2r

)1/2, where

Wr (s) = W (s) − 1r

∫ r0 W is the demeaned Brownian motion. The 90%, 95%, 99% as-

ymptotic and finite sample critical values of the sup t-statistic were reported in Table 1of PSY for various values of r0. Interestingly, in all cases the 95% critical value is muchlarger than zero. The intuition for the two estimators is that, re is the first time when theevidence of explosive behavior is found while rf is, given an explosive subsample of datahaving been found, the first time when the evidence of explosive behavior disappears.

The t-statistics in (22) can be replaced with the coeffi cient-based statistics. Phillipsand Yu (2011) obtained the asymptotic distribution of the corresponding sup c-statistic assupr∈[r0,1] r

∫ r0 WrdW/

∫ r0 W

2r . The 90%, 95%, 99% asymptotic and finite sample critical

values of the sup c-statistic can similarly be obtained. As for the sup t-statistic, the 95%critical values of the sup c-statistic (both asymptotic and finite sample) are greater thanzero. For this reason, we recommend the use of 95% critical values in empirical studies.This is because, according to Part (c) of Theorem 5.1, if the data is generated from atrend-stationary model, the coeffi cient-based statistic, when obtained from the regression(16) with k ≥ 1, converges in probability to zero and hence is less than the critical value.

23

­8

­4

0

4

8

0

100

200

300

400

500

600

80 90 00 10 20 30 40 50 60 70 80 90 00 10

PD ratioW Y c­stat95% CV

79M 10­79M 11

17M 11­17M 12

28M 11­29M 09

55M 06­56M 04

87M 02­87M 09

96M 02­01M 06

09M 02­M 03

Figure 3: Date-stamping bubble periods in the S&P 500 price-dividend ratio based on thec-statistic from the proposed regression (16) with k = 1.

In addition, we recommend the use of the sup c-statistic because the coeffi cient-basedstatistic has better robustness properties under deterministic trends than the t-statistic infinite samples as argued earlier.

7.1 Stock market

In the first empirical study, we analyze the monthly S&P 500 stock price index-dividendratio over the period from January 1871 to December 2010. Figure 3 plots the data, thesequence of the sup c-statistics from the regression (16) with k = 1, and the sequenceof the 95% finite sample critical values of the sup c-statistic obtained from Monte Carlosimulations. Comparing Figure 3 with Figure 1 which is based on the sup t-statisticobtained by PSY from the regression (16) with k = 0, we can see that the empiricalfindings from the two procedure are qualitatively identical. That is, the same seven bubbleepisodes have been identified, namely the post long-depression period in 1880s, the greatcrash episode in 1920s, the postwar boom in 1950s, black Monday in 1987, the dot-combubble in 1990s, the 1917 stock market crash in 1910s, and the subprime mortgage crisisin 2008. The last two periods, namely the 1917 stock market crash in 1910s, and thesubprime mortgage crisis in 2008, although experiencing market downturns, continue tobe identified as bubble episodes.

Given that the proposed testing procedure in our paper is able to distinguish the ex-

24

Table 14: Empirical results based on the PSY procedure and the proposed procedure forthe S&P 500 stock price index-dividend ratio between October 2006 and March 2009.

PSY Proposed Procedurec-test t-test c-test t-test1.3439 1.0714 1.0282 0.73773

95% Finite Sample CV -0.0498 -0.0264 0.0809 0.0377

plosive process from trend-stationary processes, the results from Figure 3 suggest that all7 bubbles found by PSY (both the positive and negative bubbles) are robust to determin-istic trends, and the deterministic trend models are probably irrelevant for describing themovement of the price-dividend time series.

Although the PSY procedure and the proposed procedure flag the same bubble episodes,the identified bubble durations are not identical. In particular, the estimated bubble ter-mination date by the proposed method is earlier than that by the PSY method in 5 outof 7 cases. For example, according to our method, the bubble episode in 1920s ends inSeptember 1929, one month earlier than that based on the PSY method. When checkingthe S&P 500 stock price index-dividend ratio, we find that after a substantial period ofupward movements, the time series reaches 194.99 in September 1929 and drops to 172.53(or 13%) in October 1929, suggesting that September 1929 is a turning point. Clearly, thetermination date estimated by our method matches the turning point of the time seriesbetter.

To examine the empirical results for the data in downturn periods more closely, weapply the proposed regression (16) with k = 1 to the data over the period from October2006 to March 2009 whose time series plot is shown in Figure 2. Both the coeffi cient-basedstatistic and the t-statistic are reported in Table 14, together with 95% finite sample criticalvalues obtained under the null hypothesis of unit root. Clearly, both the PSY and ourproposed procedure have to reject the null hypothesis of unit root in favor of explosiveness.Again, since our empirical method is robust to deterministic trends, the negative bubbleidentified by PSY is not likely due to a deterministic trend. Our result reinforces theempirical conclusion drawn in PSY.

7.2 Real estate market

In the second empirical study, we analyze the data of the US real estate market, whichcontains the monthly S&P/Case-Shiller US National Home Price Index and the monthlyrent of primary residence, both over the period from January 1981 to June 2017 (438monthly observations in the full sample).6 The price-rent ratio is calculated for the sampleperiod.

For the purpose of comparison, we first apply the PSY method (based on the supc-statistic) with k = 0 to the price-rent ratio. Figure 4 plots the price-rent ratio, the

6The data is downloaded from Federal Reserve Bank of St. Louis. The series code for the home priceindex is CSUSHPISA while the code for the rent is CUSR0000SEHA.

25

­4

­2

0

2

4

6

8

10

.4

.5

.6

.7

.8

.9

1985 1990 1995 2000 2005 2010 2015

Price RentRatioPSY c­stat95% CV

1986M 05­1990M 04

1991M 03­1993M 06

1998M 04­2008M 01

2008M 04­2011M 06

Figure 4: Date-stamping bubble periods in the US home price rent ratio based on thePSY method.

sequence of PSY c-statistics which are calculated in a recursive backward manner withthe minimum window size chosen as 0.01 + 1.8/

√438 ≈ 42, and the sequence of the 95%

finite sample critical values of the sup c-statistic obtained from Monte Carlo simulations.It can be seen that four bubble episodes are identified by the PSY method, namely, May1986 to April 1990, March 1991 to June 1993, April 1998 to January 2008, and April2008 to June 2011. The first and third episodes correspond to the well-known periods ofreal estate market expansions in the US. However, some post-peak periods are includedas a part of bubble expansion. The price-rent ratio reached a peak in May 1989, whereas,the estimated termination date of the first bubble by PSY method is April 1990. Theprice-rent ratio reached another peak in March 2006, whereas, the PSY method estimatesJanuary 2008 to be the termination date of the third bubble. During the second and lastdetected bubble periods the real estate market experienced market downturns. Hence,according to PSY, these two periods must have negative bubbles.

We then apply the proposed regression (16) with k = 1 to the price-rent ratio of theUS real estate market. Figure 4 plots the price-rent ratio, the sequence of the c-statisticsfrom the proposed procedure calculated in a recursive backward manner with the sameminimum window size as in PSY method, and the sequence of the 95% finite samplecritical values obtained from Monte Carlo simulations. Only two bubble periods have beenidentified by the proposed method, namely, May 1986 to May 1989 and July 1998 to March2006. Although these two bubble periods correspond to the first and third bubble periods

26

­2

0

2

4

6

.4

.5

.6

.7

.8

.9

1985 1990 1995 2000 2005 2010 2015

Price RentRatioW Y c­stat95% CV

1986M 05­1989M 05

1998M 07­2006M 03

Figure 5: Date-stamping bubble periods in the US home price rent ratio based on theproposed regression method.

identified by the PSY method, they are shorter in the sense that both bubbles ended muchearlier. The first bubble ended in May 1989 according to the proposed method, elevenmonths earlier than that identified by the PSY method. The second bubble ended in March2006 according to the proposed method, twenty-two months earlier than that identifiedby the PSY method. As noted earlier, both May 1989 and March 2006 are two peaks ofthe time series. Clearly, the estimated termination dates synchronize the turning pointsby the proposed procedure. Moreover, the third and the last bubble periods identified byPSY method are not flagged as a negative bubble episode by the proposed method. Thisobservation indicates that the market downturns in 1991-1993 and in 2008-2011 may bebetter explained by deterministic trend models than pure AR models.

8 Conclusion

This paper is concerned about the performance of the right-tailed unit root tests againstexplosive alternative when the true DGP has a deterministic trend. It is shown that whenthere is a linear trend in DGP (upward or downward), the unit root tests based on theAR(1) regression without intercept tend to reject the null hypothesis of unit root in favorof the explosive alternative. Similarly, when there is a quadratic trend or a cubic trendin DGP (upward or downward), the unit root tests based on the AR(1) regressions withor without an intercept also tend to reject the null hypothesis of unit root in favor of the

27

Table 15: The asymptotic properties of the two unit root test statistics obtained fromdifferent regression models when data come from different DGPs.

Model (6) Model (7) Model (8) k = 1 Model (8) k = 2

DGP c-stat t-stat c-stat t-stat c-stat t-stat c-stat t-statLinear Trend 1.5 ∞ 0 0 0 0 0 0Quadratic Trend 2.5 ∞ 15/8 ∞ 0 0 0 0Cubic Trend 3.5 ∞ 28/9 ∞ 0 −∞ 0 0Explosive ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

explosive alternative. Extensive simulation studies reinforce the analytical findings.In order for the right-tailed unit root tests to be able to distinguish the trend station-

ary process from the explosive AR process, we propose a new autoregressive procedure.Asymptotic distributions for the coeffi cient-based statistic and the t-statistic under thenull hypothesis of unit root are derived. When data is generated from a trend-stationarymodel, we show that the null hypothesis of unit root will not be rejected. Whereas, whendata is generated from an explosive autoregressive process, we show that both test sta-tistics go to positive infinity, suggesting that the null hypothesis of unit root has to berejected in favor of explosiveness. Interestingly, our proposed procedure is nearly identicalto the augmented DF procedure discussed in PSY (2015a, Equation (4)) with an impor-tant distinction. That is our method requires k to be at least as large as 1, whereas inEquation (4) of PSY k is allowed to be zero. However, in the empirical implementation,PSY and many other empirical studies that applied the PSY method have always fixed kto be zero for simplicity.

We have applied our proposed method to real data. For the S&P 500 stock price index-dividend ratio over January 1871 to December 2010, our testing results are qualitativelyidentical to those found in PSY, hence, reinforcing the empirical conclusion of negativebubbles made in PSY, and suggesting that deterministic trend models are not relevantfor describing the movement of the price-dividend time series. However, for the price-rentratio of the US real estate market from January 1981 to June 2017, our testing resultsare different from those obtained by using PSY method. Two negative bubbles identifiedby PSY method are not flagged as bubbles any more. In addition, the termination datesof two identified positive bubble periods are estimated to be much earlier than thoseestimated by PSY method.

APPENDIX

Before we prove the asymptotic results reported in the paper, we first summarize them.Table 15 summarizes the asymptotic properties of the two unit root test statistics obtainedfrom different regression models, (6)-(8) with k = 1, 2, and when data come from differentDGPs (the linear trend model (5), the quadratic trend model (10), the cubic trend model(13), and the mildly explosive model).

28

A Proof of theorems in Section 2

Lemma A.1 Let ut = C (L) εt =∑∞

j=0 cjεt−j, where∑∞

j=0 j |cj | < ∞ and {εt} ≡i.i.d.

(0, σ2

)with finite fourth moment. Define γj = E (utut−j) for j = 0, 1, 2, . . . ,

λ = σ∑∞

j=0 cj = σC (1) , and W (·) being a standard Brownian motion. Then,(a) n−1/2

∑nt=1 ut ⇒ λW (1)

d= N

(0, λ2

);

(b) n−1∑T

t=1 ut−jutp−→ γj for j = 0, 1, 2, . . . ;

(c) n−3/2∑n

t=1 tut−j ⇒ λ(W (1)−

∫ 10 W (r) dr

)for j = 0, 1, 2, . . . ;

(d) n−5/2∑n

t=1 t2ut−j ⇒ λ

(W (1)− 2

∫ 10 rW (r) dr

)for j = 0, 1, 2, . . . ;

(e) n−(s+1)∑n

t=1 ts = 1/ (s+ 1) +O (1/n) for s = 1, 2, . . .

Proof of Lemma A.1: Proofs of these standard results are omitted.

Proof of Theorem 2.1: (a) From Model (5), we have ∆yt = δ + ∆ut, which leads to

β =

∑nt=2 yt−1yt∑nt=2 y

2t−1

= 1 +

∑nt=2 yt−1∆yt∑nt=2 y

2t−1

= 1 +

∑nt=2 yt−1 (δ + ∆ut)∑n

t=2 y2t−1

.

Based on the results in Lemma A.1, it can be obtained that

n−2n∑t=2

δyt−1 =δ

n2

n∑t=2

[δ (t− 1) + ut−1] = δ2/2 +Op(n−1

),

n−1n∑t=2

(∆ut) yt−1 =δ

n

n∑t=2

(∆ut) (t− 1) +1

n

n∑t=2

(∆ut)ut−1

=δ

n

(nun −

n∑t=2

ut

)+

1

n

(n∑t=2

utut−1 −n∑t=2

u2t−1

)= δun + γ1 − γ0 + op (1)

and

n−3n∑t=2

y2t−1 = n−3

(δ2

n∑t=2

(t− 1)2 +n∑t=2

u2t−1 + 2δ

n∑t=2

(t− 1)ut−1

)

=δ2

n3

2n3 − 3n2 + n

6+Op

(n−3/2

)= δ2/3 +Op

(n−1

).

Consequently, we have

n(β − 1

)=n−2

∑nt=2 (δ + ∆ut) yt−1

n−3∑n

t=2 y2t−1

=δ2/2 +Op

(n−1

)δ2/3 +Op (n−1)

=3

2+Op

(1

n

).

(b) Note that[se(β)]2

=(∑n

t=2 y2t−1

)−1(

1n−2

∑nt=2

(yt − βyt−1

)2). We have

n∑t=2

(yt − βyt−1

)2=

n∑t=2

[∆yt −

(β − 1

)yt−1

]2=

n∑t=2

(∆yt)2 −

(β − 1

)2n∑t=2

y2t−1,

29

which leads to [se(β)]2

=1

n− 2

[n∑t=2

(∆yt)2 /

n∑t=2

y2t−1 −

(β − 1

)2].

Note that, as n→∞,

1

n− 2

n∑t=2

(∆yt)2 =

1

n− 2

n∑t=2

(δ + ∆ut)2 =

(n− 1) δ2 +∑n

t=2 (∆ut)2 + 2δ

∑nt=2 ∆ut

n− 2

= δ2 + 2 (γ0 − γ1) + op (1) .

Together with the limits of n−3∑n

t=2 y2t−1 and n

(β − 1

)derived above, we have

n3[se(β)]2

=1

n−1

∑nt=2 (∆yt)

2

n−3∑n

t=2 y2t−1

− n3

n− 1

(β − 1

)2

p−→ δ2 + 2 (γ0 − γ1)

δ2/3−(

3

2

)2

=3δ2 + 24 (γ0 − γ1)

4δ2 ,

which leads to the final result as

tβ√n

=n(β − 1

)n3/2se

(β) p−→ (3/2)

√4δ2√

3δ2 + 24 (γ0 − γ1)=

√3 |δ|√

δ2 + 8 (γ0 − γ1).

Proof of Theorem 2.2: (a) As yt = δt+ut = δ+ yt−1 + ∆ut, the centered LS estimatortakes the form of(

α− δβ − 1

)=

( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1( ∑nt=2 ∆ut∑n

t=2 yt−1∆ut

).

Based on the limits of n−2∑n

t=2 yt−1, n−3∑n

t=2 y2t−1 and n

−1∑n

t=2 yt−1∆ut derived in theproof of Theorem 2.1, we have(

1 00 n−1

)( ∑nt=2 ∆ut∑n

t=2 yt−1∆ut

)=

[un − u1

δun + γ1 − γ0 + op (1)

],

and (n 00 n2

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(1 00 n

)=

(1 δ/2δ/2 δ2/3

)−1

+Op(n−1

)=

(4 −6/δ−6/δ 12/δ2

)+Op

(n−1

).

Hence, (n 00 n2

)(α− δβ − 1

)=

(4 −6/δ−6/δ 12/δ2

)(un − u1

δun + (γ1 − γ0)

)+ op (1)

30

which leads to the result in (a).

(b) Note that se(β)takes the form of

se(β)

=

(0 1)( ∑n

t=2 1∑n

t=2 yt−1∑nt=2 yt−1

∑nt=2 y

2t−1

)−1(01

)∑n

t=2

(yt − α− βyt−1

)2

n− 3

1/2

.

As yt = δt+ ut = δ + yt−1 + ∆ut and(α, β

)p−→ (δ, 1), it can be proved that

n−1n∑t=2

(yt − α− βyt−1

)2= n−1

n∑t=2

(∆ut)2 + op (1) = 2 (γ0 − γ1) + op (1) .

Note that (0 n2

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(0n

)=

(0 1

)(n 00 n2

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(1 00 n

)(01

)=

(0 1

)( 4 −6/δ−6/δ 12/δ2

)(01

)+Op

(n−1

).

We then have

n3[se(β)]2

=(0 n2

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(0n

)∑n

t=2

(yt − α− βyt−1

)2

n− 1

= 24 (γ0 − γ1) /δ2 + op (1) .

Together with the limit of n2(β − 1

), it is obtained that

√ntβ = n2

(β − 1

)/[n3/2se

(β)]

=−12 (γ0 − γ1) /δ2 + 6 (un + u1) /δ√

24 (γ0 − γ1) /δ2+ op (1)

which leads to the result in (b) directly.

Proof of Theorem 2.3: For simplicity, we give only the proof for the regression withk = 1. The same approach can be applied straightforwardly to prove the results in generalcases where k > 1. When k = 1, the regression (8) becomes

yt = α+ βyt−1 + ψ1∆yt−1 + et.

The fact of ∆yt−1 = δ + ∆ut−1 makes α and ψ1 not consistent, which causes diffi culty toderive the limits of β and tβ. Hence, we turn to study an alternative LS regression:

yt = α∗ + β∗yt−1 + ψ

∗1∆ut−1 + e∗t , (23)

31

where(α∗, β

∗, ψ∗1

)are the LS coeffi cients. It is easy to see that

α

β

ψ1

= D′

α∗β∗ψ∗1

and se(β)

= se(β∗)

with D =

1 0 00 1 0−δ 0 1

where se

(β)and se

(β∗)are the standard errors of β and β

∗, respectively. Therefore,

β = β∗and tβ =

β − 1

se(β) =

β∗ − 1

se(β∗) = tβ∗ ,

where tβ∗ is the t statistic associated with β∗. We now focus on the regression (23) to

study the limits of β∗and tβ∗ as n→∞.

(a) As yt = δ + yt−1 + ∆ut, the centered LS estimator of the regression (23) is

α∗ − δβ∗ − 1

ψ∗1 − 0

=

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆ut−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆ut−1∑n

t=3∆ut−1

∑n

t=3yt−1∆ut−1

∑n

t=3(∆ut−1)2

−1

∑n

t=3∆ut∑n

t=3yt−1∆ut∑n

t=3∆ut−1∆ut

.With the limit of n−1

∑n

t=3yt−1∆ut obtained in the proof of Theorem (2.1), we have

1 0 00 n−1 00 0 n−1

∑n

t=3∆ut∑n

t=3yt−1∆ut∑n

t=3∆ut−1∆ut

=

un − u2

δun + γ1 − γ0

2γ1 − γ0 − γ2

+ op (1) .

Note that

1

n

∑n

t=3yt−1∆ut−1 =

1

n

∑n

t=3[δ (t− 1) + ut−1] ∆ut−1

=δ

n

[(n− 1)uT−1 −

∑n−2

t=2ut − 2u1

]+

1

n

∑n

t=3ut−1∆ut−1

= δuT−1 + γ0 − γ1 + op (1) .

We now have

n 0 00 n2 00 0 1

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆ut−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆ut−1∑n

t=3∆ut−1

∑n

t=3yt−1∆ut−1

∑n

t=3(∆ut−1)2

−11 0 0

0 n 00 0 n

=

1 δ/2 un−1 − u2

δ/2 δ2/3 δuT−1 + γ0 − γ1

0 0 2 (γ0 − γ1)

−1

+ op (1) .

32

Consequently, we haven 0 00 n2 00 0 1

α∗ − δβ∗ − 1

ψ∗1 − 0

=

1 δ/2 un−1 − u2

δ/2 δ2/3 δuT−1 + γ0 − γ1

0 0 2 (γ0 − γ1)

−1 un − u2

δun + γ1 − γ0

2γ1 − γ0 − γ2

+ op (1)

which shows that n2(β∗ − 1

)= Op (1). Hence, n2

(β − 1

)= Op (1).

(b) Note that

1

n

∑n

t=3(e∗t )

2 =1

n

∑n

t=3

(yt − α∗ − β

∗yt−1 − ψ

∗1∆ut−1

)2

=1

n

∑n

t=3

(∆ut − (α∗ − δ)−

(β∗ − 1

)yt−1 − ψ

∗1∆ut−1

)2

=1

n

∑n

t=3

(∆ut − (α∗ − δ)−

(β∗ − 1

)yt−1 − ψ

∗1∆ut−1

)∆ut

=1

n

∑n

t=3(∆ut)

2 − ψ∗11

n

∑n

t=3∆ut−1∆ut +Op

(n−2

)=

4 (γ0 − γ1)2 − (2γ1 − γ0 − γ2)2

2 (γ0 − γ1)+ op (1)

where the second equation is from yt = δ + yt−1 + ∆ut, the third equation comes fromfirst-order conditions of LS regression, and the fourth equation is based on the asymptoticresults obtained in the proof of (a). Hence,

n3[se(β∗)]2

=(0 n2 0

)∑n

t=31

∑n

t=3yt−1

∑n

t=3∆ut−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆ut−1∑n

t=3∆ut−1

∑n

t=3yt−1∆ut−1

∑n

t=3(∆ut−1)2

−10

n0

∑n

t=3(e∗t )

2

n− 5

=12

δ2

[4 (γ0 − γ1)2 − (2γ1 − γ0 − γ2)2

2 (γ0 − γ1)

]+ op (1) .

Finally, we have

√ntβ =

√ntβ∗ =

n2(β∗ − 1

)n3/2se

(β∗) = Op (1) .

B Proof of theorems in Section 3

Proof of Theorem 3.1: (a) From Model (10), we have ∆yt = −δ + 2δt+ ∆ut. Hence,

β − 1 =

∑nt=2 yt−1∆yt∑nt=2 y

2t−1

=

∑nt=2 yt−1 (−δ + 2δt+ ∆ut)∑n

t=2 y2t−1

.

Based on the results in Lemma A.1, it can be obtained that

n−3n∑t=2

yt−1 =1

n3

n∑t=2

[δ (t− 1)2 + ut−1

]= δ/3 +Op

(n−1

),

33

n−4n∑t=2

tyt−1 =1

n4

n∑t=2

t[δ (t− 1)2 + ut−1

]= δ/4 +Op

(n−1

),

n−2n∑t=2

(∆ut) yt−1 =δ

n2

n∑t=2

(∆ut) (t− 1)2 +1

n2

n∑t=2

(∆ut)ut−1

=δ

n2

((n− 1)2 un − 2

n∑t=2

(t− 1)ut−1 +n∑t=2

ut−1

)+Op

(n−1

)= δun +Op

(n−1/2

)and

n−5n∑t=2

y2t−1 = n−5

n∑t=2

(δ2 (t− 1)4 + u2

t−1 + 2δ (t− 1)2 ut−1

)= δ2/5 +Op

(n−1

).

Consequently, we have

n(β − 1

)=n−4

∑nt=2 yt−1 (−δ + 2δt+ ∆ut)

n−5∑n

t=2 y2t−1

=2δ (δ/4) +Op

(n−1

)δ2/5 +Op (n−1)

=5

2+Op

(1

n

).

(b) Note that

n−3n∑t=2

(∆yt)2 = n−3

n∑t=2

(−δ + 2δt+ ∆ut)2 =

4δ2

n3

n∑t=2

t2 +Op(n−1

)=

4δ2

3+Op

(n−1

).

Then, we have

n3[se(β)]2

=n3

n− 2

∑nt=2

(yt − βyt−1

)2∑nt=2 y

2t−1

=n3

n− 2

∑nt=2

[∆yt −

(β − 1

)yt−1

]2∑nt=2 y

2t−1

=n3

n− 2

[∑nt=2 (∆yt)

2∑nt=2 y

2t−1

−(β − 1

)2]

=4δ2/3

δ2/5−(

5

2

)2

+Op(n−1

)=

5

12+Op

(n−1

).

As a result,

tβ√n

=n(β − 1

)n3/2se

(β) p−→ 5/2√

5/12=√

15.

Proof of Theorem 3.2: (a) From Model (10), we have yt = −δ + yt−1 + 2δt + ∆ut.Hence, the centered LS estimator takes the form of(

α+ δ

β − 1

)=

( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1( ∑nt=2 (2δt+ ∆ut)∑n

t=2 yt−1 (2δt+ ∆ut)

).

34

Based on the asymptotic results obtained in the proof of Theorem 3.1, we have(n−2 0

0 n−4

)( ∑nt=2 (2δt+ ∆ut)∑n

t=2 yt−1 (2δt+ ∆ut)

)=

[δ

δ2/2

]+Op

(n−1

),

and (n−1 0

0 n

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(n2 00 n4

)=

(1 δ/3δ/3 δ2/5

)−1

+Op(n−1

).

Hence,(n−1 0

0 n

)(α+ δ

β − 1

)=

(1 δ/3δ/3 δ2/5

)−1 [δ

δ2/2

]+Op

(n−1

)=

[3δ/815/8

]+Op

(n−1

).

(b) Let ηt = 2δt+ ∆ut = ∆yt + δ. We have

1

n3

n∑t=2

(ηt)2 = 4δ2/3 +Op

(n−1

),

1

n2

n∑t=2

ηt = δ +Op(n−1

), and

1

n4

n∑t=2

yt−1ηt = 2δ1

n4

n∑t=2

tyt−1 +Op(n−2

)= δ2/2 +Op

(n−1

),

Together with the limits of (α+ δ) /n and n(β − 1

), we have

1

n3

n∑t=2

e2t =

1

n3

n∑t=2

(yt − α− βyt−1

)2=

1

n3

n∑t=2

[ηt − (α+ δ)−

(β − 1

)yt−1

]2

=1

n3

n∑t=2

[ηt − (α+ δ)−

(β − 1

)yt−1

]ηt = δ2/48 +Op

(n−1

),

where the third equation is from the first-order conditions of LS regression. Therefore,

n3[se(β)]2

=(0 n

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(0n4

)( ∑nt=2 e

2t

n2 (n− 3)

)=

(0 1

)( 1 δ/3δ/3 δ2/5

)−1(01

)δ2

48+Op

(n−1

)=

15

64+Op

(n−1

).

Consequently, we have

tβ/√n =

n(β − 1

)n3/2se

(β) =

15/8√15/64

+Op(n−1

) p−→√

15.

Proof of Theorem 3.3: We first prove the results for the regression (8) with k = 1 indetails. Then, for simplicity, we give the outline of the proof for general case with k > 1.

(a) For Model (10), we have ∆2yt = 2δ + ∆2ut which leads to

yt = 2δ + yt−1 + ∆yt−1 + ∆2ut

35

Hence, the centered LS estimator of the regression (8) with k = 1 isα− 2δ

β − 1

ψ1 − 1

=

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−1

∑n

t=3∆2ut∑n

t=3yt−1∆2ut∑n

t=3∆yt−1∆2ut

.Note that

n−2∑n

t=3yt−1∆2ut = n−2

∑n

t=3

[δ (t− 1)2 + ut−1

]∆2ut

= n−2∑n

t=3

[δ (t− 1)2

][∆ut −∆ut−1] +Op

(n−1

)= δn−2

[(n− 1)2 ∆un − 2

∑n−1

t=2t∆ut +

∑n−1

t=3∆ut

]+Op

(n−1

)= δ∆un +Op

(n−1

)and

n−1∑n

t=3∆yt−1∆2ut = n−1

∑n

t=3(2δt− 3δ + ∆ut−1) ∆2ut

= 2δn−1∑n

t=3t∆2ut + n−1

∑n

t=3∆ut−1∆2ut +Op

(n−1

)= 2δn−1 (n∆un) + n−1

∑n

t=3∆ut−1∆2ut +Op

(n−1

)= 2δ∆un + (4γ1 − 3γ0 − γ2) + op (1) .

Hence,1 0 00 n−2 00 0 n−1

∑n

t=3∆2ut∑n

t=3yt−1∆2ut∑n

t=3∆yt−1∆2ut

=

∆un −∆u2

δ∆un2δ∆un + (4γ1 − 3γ0 − γ2)

+ op (1) .

Next, note that n−2∑n

t=3∆yt−1 = n−2 (yn−1 − y2) = δ +Op

(n−1

),

n−4∑n

t=3yt−1∆yt−1 = n−4

∑n

t=3

[δ (t− 1)2 + ut−1

][2δt− 3δ + ∆ut−1]

= 2δ2n−4∑n

t=3t3 +Op

(n−1

)= δ2/2 +Op

(n−1

)and

n−3∑n

t=3(∆yt−1)2 = n−3

∑n

t=3(2δt− 3δ + ∆ut−1)2

=4δ2

n3

n∑t=2

t2 +Op(n−1

)=

4δ2

3+Op

(n−1

).

Together with the limits of n−3∑n

t=3yt−1 and n−5

∑n

t=3y2t−1 obtained in the proof of

Theorem (3.1), we haven 0 00 n3 00 0 n2

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−11 0 0

0 n2 00 0 n

36

=

1 δ/3 δδ/3 δ2/5 δ2/2δ δ2/2 4δ2/3

−1

+Op(n−1

)As a result, we haven 0 0

0 n3 00 0 n2

α− 2δ

β − 1

ψ1 − 1

=

1 δ/3 δδ/3 δ2/5 δ2/2δ δ2/2 4δ2/3

−1 ∆un −∆u2

δ∆un2δ∆un + (4γ1 − 3γ0 − γ2)

+ op (1)

which leads to n3(β − 1

)= Op (1).

(b) Note that yt = 2δ + yt−1 + ∆yt−1 + ∆2ut. Together with the fact that α, β, andψ1 are all consistent, it can be proved that

1

n

∑n

t=3(et)

2 =1

n

∑n

t=3

(yt − α− βyt−1 − ψ1∆yt−1

)2=

1

n

∑n

t=3

(∆2ut

)2+ op (1)

= 6γ0 − 8γ1 + 2γ2 + op (1) .

As a result, we have

n5[se(β)]2

= n5(0 1 0

)∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−10

10

∑n

t=3(et)

2

n− 5

=(0 1 0

) 1 δ/3 δδ/3 δ2/5 δ2/2δ δ2/2 4δ2/3

−1010

[6γ0 − 8γ1 + 2γ2] + op (1) .

Therefore,√ntβ =

n3(β − 1

)n5/2se

(β) = Op (1) .

For the regression (8) with k = 2 :

yt = α+ βyt−1 + ψ1∆yt−1 + ψ2∆yt−2 + et,

it is confronted with the problem of perfect multi-collinearity as ∆yt−1 = 2δt−3δ+∆ut−1

and ∆yt−2 = 2δt − 5δ + ∆ut−2. Note that ∆yt−2 − ∆yt−1 + 2δ = −∆2ut−1. We nowconsider the regression

yt = α∗ + β∗yt−1 + ψ

∗1∆yt−1 + ψ

∗2

(−∆2ut−1

)+ e∗t . (24)

It can be proved thatα

β

ψ1

ψ2

= D′

α∗

β∗

ψ∗1

ψ∗2

with D =

1 0 0 00 1 0 00 0 1 02δ 0 −1 1

37

which leads to β = β∗. It is also easy to prove that tβ = tβ∗ . Note that the regression

(24) does not face the problem of perfect multi-collinearity, and includes the true DGP of

yt when(α∗ β

∗ψ∗1 ψ

∗2

)=(2δ 1 1 0

). It can be proved that n3

(β∗ − 1

)= Op (1)

and√ntβ∗ = Op (1).

The same method can be extended to prove the results in the regression (8) with k > 2.

We will still have n3(β∗ − 1

)= Op (1) and

√ntβ∗ = Op (1), but the form of the limiting

distributions may change as k varies.

C Proof of theorems in Section 4

Proof of Theorem 4.1: (a) From Model (13), we have ∆yt = δ − 3δt + 3δt2 + ∆ut.Hence,

β − 1 =

∑nt=2 yt−1∆yt∑nt=2 y

2t−1

=

∑nt=2 yt−1

(δ − 3δt+ 3δt2 + ∆ut

)∑nt=2 y

2t−1

.

Based on the results in Lemma A.1, it is obtained that

n−4n∑t=2

yt−1 =1

n4

n∑t=2

[δ (t− 1)3 + ut−1

]= δ/4 +Op

(n−1

),

n−5n∑t=2

tyt−1 =1

n5

n∑t=2

t[δ (t− 1)3 + ut−1

]= δ/5 +Op

(n−1

),

n−6n∑t=2

t2yt−1 =1

n6

n∑t=2

t2[δ (t− 1)3 + ut−1

]= δ/6 +Op

(n−1

),

n−3n∑t=2

(∆ut) yt−1 = n−3n∑t=2

(∆ut)[δ (t− 1)3 + ut−1

]= δun +Op

(n−1/2

)and

n−7n∑t=2

y2t−1 = n−7

n∑t=2

[δ (t− 1)3 + ut−1

]2= δ2/7 +Op

(n−1

).

Consequently, we have

n(β − 1

)=

(3δ)n−6∑n

t=2 t2yt−1

n−7∑n

t=2 y2t−1

=3δ (δ/6) +Op

(n−1

)δ2/7 +Op (n−1)

=7

2+Op

(1

n

).

(b) Note that

n−5n∑t=2

(∆yt)2 =

(3δ)2

n5

n∑t=2

t4 +Op(n−1

)=

9δ2

5+Op

(n−1

).

Then, we have

n3[se(β)]2

=n3

n− 2

∑nt=2

(yt − βyt−1

)2∑nt=2 y

2t−1

=n3

n− 2

∑nt=2

[∆yt −

(β − 1

)yt−1

]2∑nt=2 y

2t−1

38

=n3

n− 2

[∑nt=2 (∆yt)

2∑nt=2 y

2t−1

−(β − 1

)2]

=9δ2/5

δ2/7−(

7

2

)2

+Op(n−1

)=

7

20+Op

(n−1

).

As a result,

tβ√n

=n(β − 1

)n3/2se

(β) p−→ 7/2√

7/20=√

35.

Proof of Theorem 4.2: (a) From Model (13), we have yt = δ+yt−1 +f (t)+∆ut, wheref (t) = 3δt2 − 3δt. Hence, the centered LS estimator takes the form of(

α− δβ − 1

)=

( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1( ∑nt=2 [f (t) + ∆ut]∑n

t=2 yt−1 [f (t) + ∆ut]

).

Based on the asymptotic results obtained in the proof of Theorem 4.1, we have(n−3 0

0 n−6

)( ∑nt=2 [f (t) + ∆ut]∑n

t=2 yt−1 [f (t) + ∆ut]

)=

[δ

δ2/2

]+Op

(n−1

),

and (n−2 0

0 n

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(n3 00 n6

)=

(1 δ/4δ/4 δ2/7

)−1

+Op(n−1

).

Hence,(n−2 0

0 n

)(α− δβ − 1

)=

(1 δ/4δ/4 δ2/7

)−1 [δ

δ2/2

]+Op

(n−1

)=

[2δ/928/9

]+Op

(n−1

).

(b) Let ξt = 3δt2 − 3δt+ ∆ut = ∆yt − δ. We have

1

n5

n∑t=2

(ξt)2 = 9δ2/5 +Op

(n−1

),

1

n3

n∑t=2

ξt = δ +Op(n−1

), and

1

n6

n∑t=2

yt−1ξt = 3δ1

n6

n∑t=2

t2yt−1 +Op(n−1

)= δ2/2 +Op

(n−1

).

Together with the limits of n−2 (α− δ) and n(β − 1

)derived above, we have

1

n5

n∑t=2

e2t =

1

n5

n∑t=2

(yt − α− βyt−1

)2=

1

n5

n∑t=2

[ξt − (α− δ)−

(β − 1

)yt−1

]2

=1

n5

n∑t=2

[ξt − (α− δ)−

(β − 1

)yt−1

]ξt = δ2/45 +Op

(n−1

),

39

where the third equation is from the first-order conditions of LS regression. Therefore,

n3[se(β)]2

=(0 n

)( ∑nt=2 1

∑nt=2 yt−1∑n

t=2 yt−1∑n

t=2 y2t−1

)−1(0n6

)( ∑nt=2 e

2t

n4 (n− 3)

)=

(0 1

)( 1 δ/4δ/4 δ2/7

)−1(01

)δ2

45+Op

(n−1

)=

112

405+Op

(n−1

).

Consequently, we have

tβ/√n =

n(β − 1

)n3/2se

(β) =

28/9√112/405

+Op(n−1

) p−→√

35.

Proof of Theorem 4.3: (a) From Model (13), we have ∆2yt = 6δ (t− 1) + ∆2ut, whichleads to

yt = −6δ + yt−1 + ∆yt−1 + ωt with ωt = 6δt+ ∆2ut.

Hence, the centered LS estimator of the regression (8) with k = 1 is

α+ 6δ

β − 1

ψ1 − 1

=

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−1

∑n

t=3ωt∑n

t=3yt−1ωt∑n

t=3∆yt−1ωt

.Note that

n−2∑n

t=3ωt = n−2

∑n

t=3

(6δt+ ∆2ut

)= 3δ +Op

(n−1

)n−5

∑n

t=3yt−1ωt = n−5

∑n

t=3yt−1

(6δt+ ∆2ut

)= 6δn−5

∑n

t=3yt−1t+Op

(n−2

)= 6δ2/5 +Op

(n−1

)and

n−4∑n

t=3∆yt−1ωt = n−4

∑n

t=3

[4δ − 3δt+ 3δ (t− 1)2 + ∆ut−1

] [6δt+ ∆2ut

]= 18δ2n−4

∑n

t=3t (t− 1)2 +Op

(n−1

)= 9δ2/2 +Op

(n−1

)Hence, n−2 0 0

0 n−5 00 0 n−4

∑n

t=3ωt∑n

t=3yt−1ωt∑n

t=3∆yt−1ωt

=

3δ6δ2/59δ2/2

+Op (1) .

It can also be proved that n−3∑n

t=3∆yt−1 = δ + Op

(n−1

), n−6

∑n

t=3yt−1∆yt−1 =

δ2/2 + Op(n−1

), and n−5

∑n

t=3(∆yt−1)2 = 9δ2/5 + Op

(n−1

). Together with the limits

40

of n−4∑n

t=3yt−1 and n−7

∑n

t=3y2t−1 obtained in the proof of Theorem (4.1), we have

n−1 0 00 n2 00 0 n

∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−1n2 0 0

0 n5 00 0 n4

=

1 δ/4 δδ/4 δ2/7 δ2/2δ δ2/2 9δ2/5

−1

+Op(n−1

)As a result, we haven−1 0 0

0 n2 00 0 n

α+ 6δ

β − 1

ψ1 − 1

=

1 δ/4 δδ/4 δ2/7 δ2/2δ δ2/2 9δ2/5

−1 3δ6δ2/59δ2/2

+Op(n−1

)which leads to the result of n2

(β − 1

)= −8.4 +Op

(n−1

).

To obtain the limit of tβ, we first have

n−3∑n

t=3(et)

2

= n−3∑n

t=3

(yt − α− βyt−1 − ψ1∆yt−1

)2= n−3

∑n

t=3

(ωt − (α+ 6δ)−

(β − 1

)yt−1 −

(ψ1 − 1

)∆yt−1

)2= n−3

∑n

t=3

(ωt − (α+ 6δ)−

(β − 1

)yt−1 −

(ψ1 − 1

)∆yt−1

)ωt

= n−3∑n

t=3ω2t −

(n−1 (α+ 6δ) n2

(β − 1

)n(ψ1 − 1

))n−2

∑n

t=3ωt

n−5∑n

t=3yt−1ωt

n−4∑n

t=3∆yt−1ωt

= 12δ2 −

3δ6δ2/59δ2/2

′ 1 δ/4 δδ/4 δ2/7 δ2/2δ δ2/2 9δ2/5

′−1 3δ6δ2/59δ2/2

+Op(n−1

)= 0.03δ2 +Op

(n−1

)where the third equation is from the first-order conditions of LS regression, and the fiveequation comes from n−3

∑n

t=3ω2t = n−3

∑n

t=3

(6δt+ ∆2ut

)2= 12δ2 + Op

(n−1

). As a

result, we have

n5[se(β)]2

=(0 n2 0

)∑n

t=31

∑n

t=3yt−1

∑n

t=3∆yt−1∑n

t=3yt−1

∑n

t=3y2t−1

∑n

t=3yt−1∆yt−1∑n

t=3∆yt−1

∑n

t=3yt−1∆yt−1

∑n

t=3(∆yt−1)2

−1 0

n5

0

∑n

t=3(et)

2

n2 (n− 5)

41

=(0 1 0

) 1 δ/4 δδ/4 δ2/7 δ2/2δ δ2/2 9δ2/5

−1010

(0.03δ2)

+Op(n−1

)=

21× 64

100+Op

(n−1

).

Therefore,

tβ/√n =

n2(β − 1

)n5/2se

(β) p−→ −

√21/2.

(b) Considering the regression (8) with k = 2 :

yt = α+ βyt−1 + ψ1∆yt−1 + ψ2∆yt−2 + et,

it is confronted with the problem of perfect multi-collinearity as∆yt−j = δ[3 (t− j)2 − 3 (t− j) + 1

]+

∆ut−j , for j = 1, 2. From Model (13), it can be seen that ∆2yt = 6δ (t− 1) + ∆2ut whichleads to

yt = yt−1 + ∆yt−1 + 6δ (t− 1) + ∆2ut

= 6δ + yt−1 + ∆yt−1 + 6δ (t− 2) + ∆2ut−1 + ∆2ut −∆2ut−1

= 6δ + yt−1 + ∆yt−1 + ∆2yt−1 + ∆3ut

where ∆3ut = ∆2ut −∆2ut−1. We now consider the regression

yt = α∗ + β∗yt−1 + ψ

∗1∆yt−1 + ψ

∗2∆2yt−1 + e∗t . (25)

It can be proved thatα

β

ψ1

ψ2

= D′

α∗

β∗

ψ∗1

ψ∗2

with D =

1 0 0 00 1 0 00 0 1 00 0 1 −1

which leads to β = β

∗. It is also easy to prove that tβ = tβ∗ . Note that the regression

(25) does not face the problem of perfect multi-collinearity, and includes the true DGP of

yt when(α∗ β

∗ψ∗1 ψ

∗2

)=(6δ 1 1 1

). It can be proved that n4

(β∗ − 1

)= Op (1)

and√ntβ∗ = Op (1). We omit the details for simplicity.

The same method can be extended to prove the results in the regression (8) with k > 2.

We will still have n4(β∗ − 1

)= Op (1) and

√ntβ∗ = Op (1), but the form of the limiting

distributions may change as k varies.

D Proof of theorems in Section 5

Proof of Theorem 5.1: (a) We only give the proof for the regression (16) with k = 1.It can be extended straightforwardly to the regression with k ≥ 1.

42

When the true DGP is yt = yt−1 + εt with εt ∼ iid(0, σ2

), and the regression (16)

with k = 1 is done, the centered LS estimator is α− 0

β − 1

ψ1 − 0

=

n∑t=2

1 yt−1 εt−1

yt−1 y2t−1 yt−1εt−1

εt−1 yt−1εt−1 ε2t−1

−1 ∑nt=2 εt∑n

t=2 yt−1εt∑nt=2 εt−1εt

Based on the results in Lemma (A.1) and the large sample theory for unit root processdeveloped in the literature (see, for example, Phillips (1987) and Phillips and Perron(1989)), it can be proved thatn−1/2 0 0

0 n−1 00 0 n−1

∑nt=2 εt∑n

t=2 yt−1εt∑nt=2 εt−1εt

⇒ σW (1)

σ2∫ 1

0 W (r) dW (r)0

and n1/2 0 0

0 n 00 0 1

n∑t=2

1 yt−1 εt−1

yt−1 y2t−1 yt−1εt−1

εt−1 yt−1εt−1 ε2t−1

−1n1/2 0 00 n 00 0 n

⇒

1 σ∫ 1

0 W (r) dr σW (1)

σ∫ 1

0 W (r) dr σ2∫ 1

0 [W (r)]2 dr σ2∫ 1

0 W (r) dW (r) + σ2

0 0 σ2

−1

≡ Π−1.

Hence, √n (α− 0)

n(β − 1

)(ψ1 − 0

)⇒ Π−1

σW (1)

σ2∫ 1

0 W (r) dW (r)0

,

which leads to(√n (α− 0)

n(β − 1

) )⇒ (1 σ

∫ 10 W (r) dr

σ∫ 1

0 W (r) dr σ2∫ 1

0 [W (r)]2 dr

)−1(σW (1)

σ2∫ 1

0 W (r) dW (r)

)and

n(β − 1

)⇒∫ 1

0 W (r) dW (r)−W (1)∫ 1

0 W (r) dr∫ 10 [W (r)]2 dr −

(∫ 10 W (r) dr

)2 .

Considering that the regression (16) with k = 1 covers the true DGP and all parametersare consistently estimated, it can be shown that

1

n− 4

n∑t=2

e2t =

1

n− 4

n∑t=2

(yt − α− βyt−1 − ψ1∆yt−1

)2=

1

n− 4

n∑t=2

ε2t + op (1)

p−→ σ2.

43

Consequently,

n2[se(β)]2

=[0 n 0

] n∑t=2

1 yt−1 εt−1

yt−1 y2t−1 yt−1εt−1

εt−1 yt−1εt−1 ε2t−1

−1 0n0

∑nt=2 e

2t

n− 4

⇒[∫ 1

0[W (r)]2 dr −

(∫ 1

0W (r) dr

)2]−1

.

Together with the limit of n(β − 1

), we have

tβ =n(β − 1

)n[se(β)] ⇒ ∫ 1

0 W (r) dW (r)−W (1)∫ 1

0 W (r) dr{∫ 10 [W (r)]2 dr −

(∫ 10 W (r) dr

)2}1/2

.

The results in Part (b) are the standard results of augmented Dickey-Fuller tests asgiven in Dickey and Fuller (1979). The results in Part (c) have already been obtained inTheorem (2.3), Theorem (3.3) and Theorem (4.3).

(d) For simplicity, we only prove the limits of n(β − 1

)and tβ =

(β − 1

)/se

(β)based

on the regression (16) with k = 1. The results for the regression with k > 1 can beproved similarly. We also assume ut = εt ∼ iid

(0, σ2

). With the use of the large sample

theory of mildly explosive process with serially dependent errors developed in Phillips andMagdalinos (2005) and Magdalinos (2012), the same approach can be straightforwardlyapplied to prove the results when ut is a weakly stationary process.

Given yt−1 = ρnyt−2 +ut−1 with ρn = 1+c/nθ, Phillips and Magdalinos (2007) showedyn = Op

(ρnnn

θ/2). We then have

∆yt−1 = yt−1 − ρ−1n (yt−1 − ut−1) =

ρn − 1

ρnyt−1 + ρ−1

n ut−1 =c

nθρnyt−1 + ρ−1

n ut−1

where the first term dominates the second term when t is large. Hence, the regression(16) with k = 1 faces the problem of perfect multi-collinearity. We turn to consider thetransformed regression as

yt = α∗ + β∗yt−1 + ψ

∗1

(ρ−1n ut−1

)+ e∗t , (26)

whose centered LS estimators have the following relationship with the centered LS esti-mators of the regression (16) with k = 1 : α− 0

β − ρnψ1 − 0

= D′

α∗ − 0

β∗ − ρnψ∗1 − 0

with D =

1 0 00 1 00 − c

nθρn1

,

which leads toβ − ρn =

(β∗ − ρn

)− c

nθρn

(ψ∗1 − 0

).

44

Note that α∗ − 0

β∗ − ρnψ∗1 − 0

=

n∑t=2

1 yt−1 ρ−1n ut−1

yt−1 y2t−1 yt−1

(ρ−1n ut−1

)(ρ−1n ut−1

)yt−1

(ρ−1n ut−1

) (ρ−1n ut−1

)2−1 ∑n

t=2 ut∑nt=2 yt−1ut∑n

t=2

(ρ−1n ut−1

)ut

.With the assumption of ut = εt ∼ iid

(0, σ2

), Phillips and Magdalinos (2007) proved that

n−2θρ−2nn

n∑t=2

y2t−1 ⇒

η2

4c2and n−θρ−nn

n∑t=2

yt−1ut ⇒ηξ

2c

where η and ξ are two independent and standard normally distributed random variables.We then have

n−θρ−nn

n∑t=2

yt−1

(ρ−1n ut−1

)= n−θρ−nn

(n∑t=2

yt−2ut−1 + ρ−1n

n∑t=2

ut−2ut−1

)

= n−θρ−nn

(n∑t=2

yt−1ut + y0u1 − yn−1un + ρ−1n

n∑t=2

ut−2ut−1

)

= n−θρ−nn

n∑t=2

yt−1ut + op (1)⇒ ηξ

2c

Together with the result in Wang and Yu (2016) that

n−3θ/2ρ−nn

n∑t=2

yt−1 ⇒η

c√

2c,

we now haven−1/2 0 00 n−θρ−nn 00 0 n−1

∑nt=2 ut∑n

t=2 yt−1ut∑nt=2

(ρ−1n ut−1

)ut

⇒σW (1)ηξ/ (2c)

0

,

and n1/2 0 00 nθρnn 00 0 1

n∑t=2

1 yt−1 ρ−1n ut−1

yt−1 y2t−1 yt−1

(ρ−1n ut−1

)(ρ−1n ut−1

)yt−1

(ρ−1n ut−1

) (ρ−1n ut−1

)2−1n1/2 0 0

0 nθρnn 00 0 n

⇒

1 0 σW (1)0 η2/

(4c2)

ηξ/ (2c)0 0 σ2

−1

As a result,n1/2 0 00 nθρnn 00 0 1

α∗ − 0

β∗ − ρnψ∗1 − 0

⇒1 0 σW (1)

0 η2/(4c2)

ηξ/ (2c)0 0 σ2

−1σW (1)ηξ/ (2c)

0

45

which leads to

nθρnn

(β∗ − ρn

)⇒ 2c

ξ

ηand ψ

∗1

p−→ 0.

Consequently, we have

n(β − 1

)= n

(β − ρn

)+ n (ρn − 1)

= n(β∗ − ρn

)− nc

nθρn

(ψ∗1 − 0

)+ n (ρn − 1)

= n(β∗ − ρn

)− nc

nθρn

(ψ∗1 − 0

)+ n

c

nθ

= n(β∗ − ρn

)+nc

nθ

(1− ρ−1

n ψ∗1

)−→ +∞

where the last limit comes from the fact that

n(β∗ − ρn

)= Op

(n

nθρnn

)= op (1) and

(1− ρ−1

n ψ∗1

)p−→ 1.

To prove the limit of tβ =(β − 1

)/se

(β), we first study the limit of se

(β). It is easy

to prove that

1

n

n∑t=2

(et)2 =

1

n

n∑t=2

[yt − α− βyt−1 − ψ1∆yt−1

]2=

1

n

n∑t=2

[yt − α∗ − β

∗yt−1 − ψ

∗1

(ρ−1n ut−1

)]2=

1

n

n∑t=2

(e∗t )2

Given that the true DGP of yt is covered by the transformed regression (26) and the LSestimates of the parameters are consistent, it can be shown that

1

n

n∑t=2

(e∗t )2 =

1

n

n∑t=2

(ut)2 + op (1)

p−→ σ2.

Note that

n2θ[0 1 0

] n∑t=2

1 yt−1 ∆yt−1

yt−1 y2t−1 yt−1∆yt−1

∆yt−1 yt−1∆yt−1 (∆yt−1)2

−1 010

= n2θ

[0 1 0

]D′

n∑t=2

D

1 yt−1 ∆yt−1

yt−1 y2t−1 yt−1∆yt−1

∆yt−1 yt−1∆yt−1 (∆yt−1)2

D′

−1

D

010

= n2θ

[0 1 − c

nθρn

] n∑t=2

1 yt−1 ρ−1n ut−1

yt−1 y2t−1 yt−1

(ρ−1n ut−1

)(ρ−1n ut−1

)yt−1

(ρ−1n ut−1

) (ρ−1n ut−1

)2−1 0

1− cnθρn

= n2θ

[0 1 − c

nθρn

]n−1/2 0 00 n−θρ−nn 00 0 1

1 0 σW (1)

0 η2/(4c2)

ηξ/ (2c)0 0 σ2

−1

+ op (1)

46

×

n−1/2 0 00 n−θρ−nn 00 0 n−1

01

− cnθρn

= n2θ

[0 n−θρ−nn − c

nθρn

]1 0 σW (1)

0 η2/(4c2)

ηξ/ (2c)0 0 σ2

−1

+ op (1)

0n−θρ−nn− cn1+θρn

=[0 ρ−nn − c

ρn

]1 0 σW (1)

0 η2/(4c2)

ηξ/ (2c)0 0 σ2

−1

+ op (1)

0ρ−nn− cnρn

p−→

[0 0 −c

]1 0 σW (1)0 η2/

(4c2)

ηξ/ (2c)0 0 σ2

−1 000

= 0

Hence, we have

n2θ[se(β)]2

= n2θ[0 1 0

] n∑t=2

1 yt−1 ∆yt−1

yt−1 y2t−1 yt−1∆yt−1

∆yt−1 yt−1∆yt−1 (∆yt−1)2

−1 010

∑nt=2 (e)2

n− 4

p−→ 0.

Therefore, as n→∞,

tβ =β − 1

se(β) =

nθ(β − 1

)nθse

(β) =

nθ(β − ρn

)+ nθ (ρn − 1)

nθse(β)

=op (1) + c

op (1)

p−→ +∞.

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